What is the integral of ((x+2)^2)/(x^2+4) ?

The integral of ((x+2)^2)/(x^2+4) is 2ln|x^2+4|+x+2+C

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integral of ((x+2)^2)/(x^2+4)

\int \frac{( x + 2 ) ^{2}}{x ^{2} + 4} d x

2 ln x^{2} + 4 + x + 2 + C

Solution steps

\int \frac{( x + 2 ) ^{2}}{x ^{2} + 4} d x

Apply u-substitution

: \int \frac{u ^{2}}{u ^{2} - 4 u + 8} d u

= \int \frac{u ^{2}}{u ^{2} - 4 u + 8} d u

\frac{u ^{2}}{u ^{2} - 4 u + 8} = \frac{4 u - 8}{u ^{2} - 4 u + 8} + 1

= \int \frac{4 u - 8}{u ^{2} - 4 u + 8} + 1 d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{4 u - 8}{u ^{2} - 4 u + 8} d u + \int 1 d u

\int \frac{4 u - 8}{u ^{2} - 4 u + 8} d u = 2 ln u^{2} - 4 u + 8

\int 1 d u = u

= 2 ln u^{2} - 4 u + 8 + u

Substitute back

u = x + 2

= 2 ln (x + 2)^{2} - 4 (x + 2) + 8 + x + 2

Expand

(x + 2)^{2} - 4 (x + 2) + 8 : x^{2} + 4

= 2 ln x^{2} + 4 + x + 2

Add a constant to the solution

= 2 ln x^{2} + 4 + x + 2 + C

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What is the integral of ((x+2)^2)/(x^2+4) ?
The integral of ((x+2)^2)/(x^2+4) is 2ln|x^2+4|+x+2+C