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

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

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integral of (In(x+1))/(sqrt(x+1))

\int \frac{I n ( x + 1 )}{x + 1} d x

I n \frac{2}{3} (1 + x)^{\frac{3}{2}} + C

Solution steps

\int \frac{I n ( x + 1 )}{x + 1} d x

Take the constant out:

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

= I n \cdot \int \frac{1 + x}{1 + x} d x

Simplify

\frac{1 + x}{1 + x} : 1 + x

= I n \cdot \int 1 + x d x

Apply u-substitution

: \int u d u

= I n \cdot \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= I n \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 1 + x

= I n \frac{2}{3} (1 + x)^{\frac{3}{2}}

Add a constant to the solution

= I n \frac{2}{3} (1 + x)^{\frac{3}{2}} + C

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