







Dividing a region in half 
20180224 

From mandy: There is a line through the origin that divides the region bounded by the parabola y=4x−5x2y=4x−5x2 and the xaxis into two regions with equal area. What is the slope of that line? Answered by Penny Nom. 





Integrate (x^2  4x + 4) ^4/3 
20160915 

From Ifah: Hi i have questions please answer
Integral 2 sampai 3 dari (x²  4x + 4) ^4/3 dx Answered by Penny Nom. 





The integral of a sum 
20160310 

From Rahul: How to solve definite integral of a sum. The specific problem is as follows,
Integral of ( 1+ sum of x^k, k=1 to k=n), x=0 to x=b *dx.
The answer is b + sum of b^(k)/k, k=2, to k=n+1. I understand only the integral
of first term. But integral of the sum I do not understand at all. Answered by Penny Nom. 





Integration of dx/(x^2+1)^3 
20160107 

From Ishank: Integration of dx/(x^2+1)^3 Answered by Penny Nom. 





The area bounded by the Xaxis and y=x^(2)4 from 5 to 0 
20140415 

From Lexie: Determine the area that is bounded by the following curve and the xaxis on the interval below. (Round your answer to three decimal places)
y=x^(2)4, 5 ≤ x ≤ 0
The answer is 32.333 but I have no idea how to get there. Answered by Penny Nom. 





Integration from 0 to 2pi of 1/(3cos x + 2) dx 
20130204 

From ankit: Integration from 0 to 2pi of 1/(3cos x + 2) dx Answered by Harley Weston. 





An integral 
20121216 

From Slavena: integration of (lnx)^2 / x dx Answered by Penny Nom. 





Integral 1/(25x^2)^3/2 
20120222 

From John: Integral 1/(25x^2)^3/2 Answered by Harley Weston. 





A volume of revolution 
20120111 

From john: find volume of solid generated by revolving the region in the first quadrant bounded by the curve y squared=x cubed, the line x=4 and the xaxis about the line y=8. The answer in the back of the book is 704 pi divided by5 Answered by Penny Nom. 





An antiderivative of the square root of (8t + 3) 
20110419 

From Caitlyn: I know how to take an antiderivative. But this one's stumping me. I need it to finish a problem.
What's the antiderivative of the square root of (8t + 3)
~Caitlyn= Answered by Penny Nom. 





Integrating ln^3x/x 
20110114 

From ken: y=ln^3x/x from x=1 to x=11 Answered by Penny Nom. 





The integral of (x^2*exp(x)/(exp(x)1)^2 
20100809 

From sujoy: please find this integral for me
int(x^2*exp(x)/(exp(x)1)^2 Answered by Robert Dawson. 





Integration of sin^3 (2x) 
20100529 

From ascher: how do you integrate this equation
∫ sin^3 (2x) dx Answered by Robert Dawson and Penny Nom. 





Integrate the ((4th root of x^3)+1) dx 
20100412 

From Bridget: integrate the ((4th root of x^3)+1) dx Answered by Tyler Wood. 





The integral of X^3/the square root of 1x^2 dx 
20100307 

From William: The integral of X^3/the square root of 1x^2 dx. Answered by Harley Weston. 





Integration of cos (x^2+y^2) dx dy 
20100212 

From anna: How do you evaluate the integration of integration of cos (x^2+y^2) dx dy? Answered by Tyler Wood. 





An integral 
20100129 

From amr: find the value of the following integral:
(1+x)power (1/2) dx Answered by Robert Dawson. 





The integral of sqrt(x) * ln(x) 
20091012 

From Oana: Hi! Can you tell me please how much is : ∫ (√x * ln x) ? Answered by Penny Nom. 





Integration of cos (x^2) dx 
20091012 

From Venkat: How do you evaluate the integration of cos (x^2) dx ? Answered by Harley Weston. 





solve integral of ( x^2+x+1)^5 
20090918 

From jaka: solve integral of ( x^2+x+1)^5 Answered by Robert Dawson. 





An antiderivative problem 
20090813 

From Indrajit: ∫4e^x + 6e^x/(9e^x + 4e^x)dx = Ax + Bloge(9e2x  4) + C
then A=?......B=?.....C=?
plz solve it...."^" stands for "to the power of".... Answered by Harley Weston. 





The integral of x^x 
20090618 

From ANGIKAR: what would be the integration of (X^Xdx)?
give answer in details. Answered by Robert Dawson and Harley Weston. 





The integral of a to power x squared 
20090428 

From JIM: WHEN I ATTENDED U.OF T. (TORONTO ) MANY YEARS AGO
WE WERE TOLD THE FOLLOWING INTEGRAL COULD NOT BE
SOLVED : a to power x squared . is this still true ?
CURIOUS , JIM Answered by Robert Dawson. 





The integral of the square root of the sine function 
20090407 

From Indrajit: how to integrate this derivative???
∫√sinx Answered by Harley Weston. 





The area between the xaxis and a curve 
20090218 

From Lauren: This is from a homework question I can't figure out.
Let R be the region in the fourth quadrant enclosed by the xaxis
and the curve y= x^2  2kx, where k > 0. If the area of the region R is 36
then what is the value of k? Answered by Robert Dawson. 





The integral of f(x)dx and the integral of f(x4)dx 
20090102 

From Katie: Calculus:
If the definite integral from 2 to 6 of f(x)dx=10 and the definite integral from 2 to 6 of f(x)dx=3, then the definite integral from 2 to 6 of f(x4)dx= ?
I don't understand how to solve definite integrals when the function has something more than just x inside the parenthesis such as f(4x). Answered by Robert Dawson. 





Antiderivative of 1/(x(1  x)) 
20081022 

From Matt: derivative of dx/(x(1x))
From what I've seen I should break apart the equation as such
derivative of dx/x  dx/(1x)
and then get the 2 corresponding log functions.
If that is correct why does this factoring work, if that is incorrect what is the proper way to find the derivative. Answered by Harley Weston. 





Integration 
20081017 

From hazela: integration of
x sinx cosnx Answered by Harley Weston. 





Integral of X^2 
20080728 

From Hemanshu: Integral of X^2 Answered by Janice Cotcher. 





The volume of a solid 
20080604 

From Tom: The base of a certain solid is a circle of radius 2 while cross sections perpendicular to the
base are isosceles triangles of height 1. What is the volume of the solid? Answered by Harley Weston. 





The weight of a concrete column 
20080511 

From russell: a cylindrical form is filled with a slow curing concrete. The base of the form
is 10 ft in radius, and height is 25 ft. while the concrete hardens, gravity
causes the density to vary from a density of 90 lbs/ft^3 at the bottom to a
density of 50 lb/ft^3 at the top. Assume that the density varies linearly
from the top to the bottom, and compute the total weight of the resulting
concrete column Answered by Harley Weston. 





Integrate 2x(x^21)^5 dx 
20080417 

From joey: pls help me to integrate
2x(x^21)^5 dx using the substitution u=x^21 Answered by Harley Weston. 





The integral of dx / (4x^2  25)^3/2 
20080401 

From Meghan: I have a question from the trigonometric substitution of my calculus course.
integral of dx / (4x^2  25)^3/2 Answered by Harley Weston. 





Integrate (4+x^2) / (x (2+x)^2) 
20080225 

From Russell: integrate (4+x^2) / (x (2+x)^2)
I really need to know the steps involved in doing this integral, I already have
the answer through wolfram integrator but for some reason can not
get to the answer given. Please help Answered by Harley Weston. 





Two regions with equal area 
20080213 

From James: There is a line through the origin that divides the region bounded by the parabola y=3x5x^2 and the xaxis into two regions with equal area. What is the slope of that line? Answered by Harley Weston. 





Integration by parts 
20080130 

From seth: hi i really dont understand integr
ation by parts. for example, the integral(t^2sintdt. i have u=t^2 and v'=sint also u'=t^/3 v=cost
for the formula i have uvintegralvu' dx this is all well and good but i cant get it right. Answered by Harley Weston. 





The integral of Sqrt(sin x) 
20080124 

From tariq: find integration
Sqrt(sinx) dx , from x=bi/4 to 3bi/4 Answered by Harley Weston. 





An integral 
20080124 

From Meghan: Evaluate the indefinite integral of (x^3 +1)^(1/3) * x^5 * dx. Answered by Brennan Yaremko. 





Integration 
20080123 

From Russell: how to integrate
1/x ln^2 [x] dx Answered by Harley Weston. 





Two integrals 
20071109 

From Akilan: how to integrate these (tan x)^6(sec x)^4 and sinh(x)(cosh(x))^2.
Please send me how to do this question. Having exams on Monday. Please help. Answered by Harley Weston. 





Integration 
20070202 

From Jenna: How do I integrate ( ln(x) ) / ( x^2 ) ? Answered by Penny Nom and Claude Tardif. 





The volume of a frustum of a pyramid 
20070117 

From Sam: Find the volume of a frustum of a pyramid with square base of side b, square top of side a, and height h. Answered by Penny Nom. 





Integrate x^8 (x^8 + 2)^2 ((x^8 + 2)^3 + 1)^4 
20070109 

From James: How do you integrate x^8 (x^8 + 2)^2 ((x^8 + 2)^3 + 1)^4 Answered by Penny Nom. 





An Integral 
20060730 

From Aniket:
I am Aniket studing in 12 th standard At Mumbai
I have following integration problem please give me a solution
integral of 1/under root of (5x^{2}  2x) dx
Answered by Penny Nom. 





The integral of tan^4x 
20050331 

From Nicole: college student. need help with tough integral.
the integral of 2tan^{4}xdx
Answered by Penny Nom. 





Approximating an integral 
20050127 

From A student: i used excel to find the area under a graph instead of integrating the function the analytical way. These are the different answers i got with different width of the xvalues, the smaller the xincrement gets the more precise the area is when compared to the answer obtained the numerical way. I have trouble seeing why the answers i get is getting bigger and bigger and not smaller can someone plz explain? Answered by Penny Nom. 





Integrating e^sin(x) 
20040804 

From A student: I need to know that how to solve the integral " e^sin x", Answered by Harley Weston. 





A helicopter rises vertically 
20030902 

From Kate: A helicopter rises vertically and t seconds after leaving hte ground its velocity is given in feet per second by v(t) = 8t + 40 / (t+2)^{2} How far above the ground will the helicopter be after 3 seconds? Answered by Penny Nom. 





Odd powers of sine and cosine 
20030625 

From Antonio: Can you please tell me how to integrate a trig function involving sine and cosine? I know if the powers of both the sine and cosine are even and nonnegative, then I can make repeated use of the powerreducing formulas. But for the question I have on my hand, the powers of both sine and cosine are odd: ( sin3x + cos7x ) dx. Answered by Harley Weston. 





Integrating e^x sin(x) 
20030503 

From Lech: I am having trouble integrating the following expression by parts: ex sin(x) I used the integrator at http://www.integrals.com/ to find the solution, ? 1/2 ex cos(x) + 1/2 ex sin(x). This is easy to confirm by differentiation, however I am confounded as how to arrive at the answer. Answered by Penny Nom and Claude Tardif. 





Integration of 1/(2+cos(x)) 
20030107 

From A student: integral from pi to 0 of dx/(2+cos x) i used the substitution t=tan(x/2) and i ended up with integral from +infinity to 0 of 2dt/(t^{2}+3) which looks like an inverse tan function , and i ended up with sqr(27)/2 pi , which is not the same as my calculator's answer , so i suspct i am doing some thing wrong. can some one tell me where i am going wrong please. Answered by Penny Nom. 





Integrating x^x 
20020618 

From Jeremy: I am a student at the University of Kansas and I am wondering if there is a general antiderivative for x^{ x} (i.e. the integral of x^{ x} dx)? I've looked in a bunch of Table of Integrals and have found nothing (can you guys find it?), so I'm sort of wondering if this may be a research type question. Answered by Claude Tardif. 





Arc length 
20020417 

From Vix: Find the point on the curve r(t)=(12sint)i(12cost)j+5tk at a distance 13pi units along the curve from the point (0,12,0) when t=0 in the direction opposite to the direction of increasing arc length. Answered by Harley Weston. 





The probability that the area of a triangle is smaller that 1/8 
20011221 

From Hoda: Given unit square pick two points on adjacent sides of the square , what is the probability that the area of the resulting triangle is smaller that 1/8? I tried to tackle the problem by first sketching a square with the four vertices at (0,0), (1,1), (0,1) and (1,1). The answer seems to be the area under the curve y=0.25/x between 1 and 0, but how can you integrate such a function when ln0 is undefined? any help would be highly appreciated. Answered by Andrei Volodin. 





Area between curves 
20010613 

From Phil:
question 1 find the area bound by the curves y = x^{2} + 2x + 3 and y = 2x + 4 question 2 Find the volume generated by rotating the curve x^{2} + y^{2} = 9 about the xaxis Answered by Harley Weston. 





National consumption function 
20010509 

From Brian: If consumption is $11 billion when disposable income is 0 and the marginal propensity to consume is dC/dy = 1/(2y+4)1/2+0.3(in billions of dollars), find the national consumption function. Answered by Harley Weston. 





The average value of a continuous function 
20010508 

From Esther: The average value of a continuous function y = f(x) on the interval [a,b] is given by ________________? Answered by Harley Weston. 





Oil revenue 
20010421 

From Brian: Suppose that t months from now an oil well will be producing crude oil at the rate of r(t), not a constant, barrels per month and that the price of crude oil will be p(t), not a constant, dollars per barrel. Assume that the oil is sold as soon as it is extracted from the ground.  Find an expression for the total revenue from the oil well, R(t).
 A certain oil well that currently yields 400 barrels of crude oil a month will run dry in 2 years. The price of crude oil is currently $18 per barrel and is expected to rise at a constant rate of 3 cents per barrel per month. What will be the total revenue from this well? {Hint: Model the degraded production rate with the equation:
r(t) = (ABt)e^{0.04t}} Answered by Harley Weston. 





Integration by parts 
20010409 

From A student: how do you integrate x tan^{1}x dx, i know it can be done by integration by parts maybe, but i'm not sure.... Answered by Claude Tardif and Harley Weston. 





Two integrals 
20010403 

From Jim: I'm having trouble with these integrals. Can you help me out? 1)the integral of:
x^{5} arctan x dx 2)the integral of:
2x^{5} + 9x^{4} + 19x^{3} + 13x^{2}  5x  25  dx x^{4} + 4x^{3} + 5x^{2} Answered by Claude Tardif. 





A jogger 
20010312 

From Bill: At time t=0 a jogger is running at a velocity of 300 meters per minute. The jogger is slowing down with a negative acceleration that is directly propotional to time t. This brings the jogger to a stop in 10 minutes. a) write an expression for the velocity of the jogger at time t. b) what is the total distance traveled by the jogger in that 10minute interval. Answered by Harley Weston. 





How do you integrate secant(theta)? 
20001222 

From Robert Williamson: How do you integrate secant(theta)? I know the answer is ln [sec(theta) + tan(theta)] but how do you get there? Answered by Claude tardif. 





A nonintegrable function 
20001203 

From Mark Spilker: I have a proof that I cannot do here it goes. Let F(x)= 1 if x is a rational number 0 if x is an irrational number Prove the function is not intregrable on the interval (0,1). Hint: Show that no matter how small the norm of the partition, the Riemann Sum for the SigmaNotation F(w_{i}) Delta_{i}X is not unique. Answered by Harley Weston. 





An integration problem 
20001130 

From A student: If a>0 and the integral from b to 0 of 1/(1+x) equals 1/2 the integral from a to 0 of 1/(1+x), express "b" in terms of "a" Answered by Claude tardif. 





Comparing an integral and a sum 
20001121 

From Douglas Norberg: A fellow teacher asked me about a problem she wanted to give to her students. It involved whether to take a million dollars or a penny doubled a number of times. I was able to determine the number must have been .01 * 2^{30} which is about $10 million and a lot more than $1 million. To check that I was right I used a spreadsheet and did a Riemann sum. When I finished I reasoned that I had done the task in several steps and I could have done it in 1 step. Thus I integrated .01 * 2^{x} from 0 through 30 but the number I got was $15,490,820.0324. Why the difference? Answered by Harley Weston. 





Dividing a region in half 
20000921 

From Kerry: There is a line through the origin that divides the region bounded by the parabola y=xx^{2} and the xaxis into two regions with equal area. What is the slope of the line? Answered by Penny Nom. 





Calculus Research Questions 
20000522 

From William Wright: I am a Calculus Teacher, and me and my class ran into these two problems without solutions in my manual, we got answers, but are unable to check them. If anyone gets this email and can respond to this with the solutions it be greatly appreciated. . . . Answered by Harley Weston. 





Two calculus problems 
20000501 

From Kaushal Shah: How Do WE Integrate the following Functions,  Integral xtanx dx
 How was natural base "e" discovered and why e=2.7.......
Answered by Claude Tardif. 





The area of a triangle using calculus 
20000415 

From Todd Bowie: Hi, I am not a student but am reviewing calculus for an upcoming interview. I would like to know how to derive the area of a triangle using calculus. Thanks! Answered by Patrick Maidorn. 





Some integration problems 
20000223 

From Tim Valentine: I am having a great deal of difficulty with the following integrals, can you help? I think they need the use of trig substitution or integration by parts but I cannot figure out how to begin. Thanks! The integral of 1/(2+3x^{2}) dx. and The integral of x * square root of (4x+5) dx. Answered by Harley Weston. 





Two calculus problems 
19991213 

From Alan: I have 2 questions that are very new to me, they were included on a quiz and the material was never covered. Our teacher never explained the purpose and detailed explanation of how to solve the problem. Could you help? Thanks. Question 1: A ball is falling 30 feet from a light that is 50 feet high. After 1 sec. How fast is the shadow of the ball moving towards the light post. Note that a ball moves according to the formula S=16t^2 Question 2: How many trapezoids must one use in order for the error to be less than 10^8 if we want to find the area under the curve Y=1/X from 1 to 2. Find the exact area, Graph the function and use the trap rule for the "N" that you found. Answered by Harley Weston.






The area and the circumference of a circle. 
19980827 

From Jason Wright: I was looking at the relationship of the area of a circle and the circumference when I realized that 2*pi*r is the derivative of pi*r^2. I was wondering if there is any connective deep dark meaning as to why this appears to be related. Thanks for any help you can give me! Answered by Walter Whiteley. 





Calculus problems 
19980713 

From Lorraine: I'm stuck again. Can you help? This involves integration using the method of partial fractions the integral of: 7x(to the 5th)  2x(cubed) + 3 dx  x(to the fourth)  81 Do I have to do long division to reduce the numerator to the fourth power? the integral of: 4 16x +21x(squared) + 6x(cubed)  3x(fourth) dx  x(cubed)(x  2)(squared) Lorraine Answered by Harley Weston. 





A Calculus Problem 
19980628 

From Lorraine: I'm a postsecondary student taking calculus by correspondence. I'm stuck on the following question (and similar ones) Can you help? Evaluate the following indefinite integral: d(theta)  1 + sin (theta) (It says to multiply both numerator and denominator by: 1  sin(theta) Thanks Lorraine Answered by Harley Weston. 

