Tim Valentine
12 Grade, age 18

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+3x2) dx.


The integral of x * square root of (4x+5) dx.

Hi Tim,

Part of learning to integrate is "pattern recognition". What you have to do is look at 1/(2+3x2) and see 1/(1+u2) which is the derivative of tan-1u. That is, you want the denominator to be 1 plus a square. (Actually there will be a constant multiplier in front but constant multipliers in integration problems are easily handled.)

With this in mind write the integrand as

Now use the substitution

The second integral is easier than it looks. Just use the substitution u = 4x + 5 and then x = (u - 5)/4.

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