Integration Using u-sub

Question

Is there a way to utilize u-substitution or other simple methods of integration to find the integral of 1/(xsqrt(x+1))? Note that I've done the integration using a u-substitution and partial fraction combination but I'm not supposed to know about partial fractions yet so that doesn't seem to be a legitimate method.

Answer 1

Let U = sqrt(x+1) ; Then U^2 = x+1 ---> U^2-1 = x

then dU/dx = (1/2)(x+1)^(-1/2) = 1/ [ 2 * sqrt(x+1)] --> dx = 2*Sqrt(x+1)*du

2*du replaces sqrt(x+1) in the denominator

U^2-1 repalces x in the denominator

the integral becomes

2 * dU / (U^2-1) = 2* dU / [(U+1)(U-1)]

which indeed can be done by partial fraction decomposition

perhaps a trig substitution may be alternative

Answer 2

The technique with these types of integrals is to let u = the square root expression and then solve that expression for x.

1. u = sqrt(x+1)
2. x = u²-1
3. du = 1/2sqrt(x+1) dx. --------->>> 2du = 1 / sqrt(x+1) dx

Now, the integral can be done simply:

1. ∫1/x√(x+1) dx = 2 ∫ 1/(u²-1) du = ln ( (u-1) / (u+1) )

and you can convert it back to an expression for x

This idea is common for integrals that involve a root of an expression of x that are being multiplied by some power of x.

Hope this helped.

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