What is the integral of (x^2+1)/(sqrt(x+1)) ?

The integral of (x^2+1)/(sqrt(x+1)) is 2(1/5 (x+1)^{5/2}-2/3 (x+1)^{3/2}+2sqrt(x+1))+C

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integral of (x^2+1)/(sqrt(x+1))

\int \frac{x ^{2} + 1}{x + 1} d x

2 (\frac{1}{5} (x + 1)^{\frac{5}{2}} - \frac{2}{3} (x + 1)^{\frac{3}{2}} + 2 x + 1) + C

Solution steps

\int \frac{x ^{2} + 1}{x + 1} d x

Apply u-substitution

: \int 2 (u^{4} - 2 u^{2} + 2) d u

= \int 2 (u^{4} - 2 u^{2} + 2) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int u^{4} - 2 u^{2} + 2 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

= 2 (\int u^{4} d u - \int 2 u^{2} d u + \int 2 d u)

\int u^{4} d u = \frac{u ^{5}}{5}

\int 2 u^{2} d u = \frac{2 u ^{3}}{3}

\int 2 d u = 2 u

= 2 (\frac{u ^{5}}{5} - \frac{2 u ^{3}}{3} + 2 u)

Substitute back

u = x + 1

= 2 (\frac{( x + 1 ) ^{5}}{5} - \frac{2 ( x + 1 ) ^{3}}{3} + 2 x + 1)

Simplify

2 (\frac{( x + 1 ) ^{5}}{5} - \frac{2 ( x + 1 ) ^{3}}{3} + 2 x + 1) : 2 (\frac{1}{5} (x + 1)^{\frac{5}{2}} - \frac{2}{3} (x + 1)^{\frac{3}{2}} + 2 x + 1)

= 2 (\frac{1}{5} (x + 1)^{\frac{5}{2}} - \frac{2}{3} (x + 1)^{\frac{3}{2}} + 2 x + 1)

Add a constant to the solution

= 2 (\frac{1}{5} (x + 1)^{\frac{5}{2}} - \frac{2}{3} (x + 1)^{\frac{3}{2}} + 2 x + 1) + C

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What is the integral of (x^2+1)/(sqrt(x+1)) ?
The integral of (x^2+1)/(sqrt(x+1)) is 2(1/5 (x+1)^{5/2}-2/3 (x+1)^{3/2}+2sqrt(x+1))+C