Problem to solve
Find the indefinite integral \[\int\frac{\sqrt{x+2}}{x+1} dx\]
Solution
STEP 1
Take the integral: integral sqrt(x + 2)/(x + 1) dx
STEP 2
For the integrand sqrt(x + 2)/(x + 1), substitute u = sqrt(x + 2) and du = 1/(2 sqrt(x + 2)) dx: = 2 integral u^2/(u^2 - 1) du
STEP 3
For the integrand u^2/(u^2 - 1), do long division: = 2 integral-1/(2 (u + 1)) + 1 + 1/(2 (u - 1)) du
STEP 4
Integrate the sum term by term and factor out constants: = - integral1/(u + 1) du + integral1/(u - 1) du + 2 integral1 du
STEP 5
For the integrand 1/(u + 1), substitute s = u + 1 and ds = du: = - integral1/s ds + integral1/(u - 1) du + 2 integral1 du
STEP 6
The integral of 1/s is log(s): = -log(s) + integral1/(u - 1) du + 2 integral1 du
STEP 7
For the integrand 1/(u - 1), substitute p = u - 1 and dp = du: = -log(s) + integral1/p dp + 2 integral1 du
STEP 8
The integral of 1/p is log(p): = log(p) - log(s) + 2 integral1 du
STEP 9
The integral of 1 is u: = log(p) - log(s) + 2 u + constant
STEP 10
Substitute back for p = u - 1: = -log(s) + 2 u + log(u - 1) + constant
STEP 11
Substitute back for s = u + 1: = 2 u + log(u - 1) - log(u + 1) + constant
STEP 12
Substitute back for u = sqrt(x + 2): = 2 sqrt(x + 2) + log(sqrt(x + 2) - 1) - log(sqrt(x + 2) + 1) + constant
STEP 13
An alternative form of the integral is: = 2 (sqrt(x + 2) - tanh^(-1)(sqrt(x + 2))) + constant
STEP 14
Which is equal to: Answer: | | = 2 sqrt(x + 2) - 2 tanh^(-1)(sqrt(x + 2)) + constant
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Edited by tododemate® - 2021