Hello,
I am having trouble integrating the following expression by parts:

e^x sin(x)

I used the integrator at http://www.integrals.com/ to find the solution,

– 1/2 e^x cos(x) + 1/2 e^x sin(x).

This is easy to confirm by differentiation, however I am confounded as how
to arrive at the answer.

I am beginning to get frustrated with myself; I grasp concepts quickly and
more importantly understand them (unlike my colleagues), but on many occasions when a question has a substantial twist I fall apart.
(High-school level - i think)

Thank you,
Lech

Hi Lech,

I am going to let I be the integral of e^x sin(x) and start by integrating by parts with

u₁ = sin(x) and dv₁ = e^x dx

This gives

I = u₁v₁ - J

where J is the integral of x cos(x).

Now integrate J by parts with

u₂ = cos(x) and dv₂ = e^x dx

This gives

I = u₁v₁ - [u₂v₂ + I]

Solve for I.

Cheers,
Penny

Hi Lech,

I think that ``the integrator'' uses an algorithm that is different
from what is usually seen in calculus courses: it guesses what should
be the form of the solution, and then differentiates to do the fine tuning.
It is just like guessing that the integral of x³ e^x should be of the form
a(x³ e^x) + b(x² e^x) + c(x e^x) + d(e^x), and taking the derivative
to find the constants a, b, c and d. The standard textbook method is to
hold up your fingers and ask yourself "Can I do this by substitution? Can
I do this by parts? ... '' and then further decide what to substitute for what,
or what are the parts. This is not really a well defined procedure that can be
implemented on a computer, because of all the guessing involved.

Anyway,I feel that you have grasped the main point: However difficult it is
to integrate an expression, it is always easy to verify the solution by
differentiation. It is actually the fundamental theorem of calculus that
you have fully understood, and you could be trusted to use a computer
calculus package wisely and not accept gibberish for an answer.

One of the fundamental open questions in theoretical computer science is
``If it is easy to confirm an affirmative answer to a given question,
is it easy to find such an affirmative answer?'' (See
the P vs NP problem in http://www.claymath.org/Millennium_Prize_Problems/).
I feel that it also has its counterpart in mathematical education in the form
"What is the role of computers in calculus courses?'': We could
decide to remove integration by parts, partial fractions, or other methods
from the curriculum and use a calculus package instead. But if the students
still check their answer to the integral of e^x sin(x) by looking at the
back of the book instead of confirming their answers by differentiating,
I think that not much is gained.

Claude

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