Example of integrals involving root - integration technique, Mathematics

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Evaluate the following integral.

∫ (x+2 / 3√(x-3)) (dx)

Solution

Occasionally while faced with an integral that consists of a root we can make use of the following substitution to make simpler the integral into a form which can be easily worked with.

u = 3√(x - 3)

Thus, in place of letting u be the stuff under the radical as we frequently did in Calculus I we let u be the whole radical. Here, there will be a little more work here as we will as well need to know what x is thus we can substitute in for that in the numerator and thus we can compute the differential, dx. Though, this is easy enough to get.

Simply solve the substitution for x as follows,

 x = u3 + 3

dx = 3u2 du

By using this substitution the integral is now,

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