What is the integral of 3/(2sqrt(x)(1+x)) ?

The integral of 3/(2sqrt(x)(1+x)) is 3arctan(sqrt(x))+C

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integral of 3/(2sqrt(x)(1+x))

\int \frac{3}{2 x ( 1 + x )} d x

3 arctan (x) + C

Solution steps

\int \frac{3}{2 x ( 1 + x )} d x

Take the constant out:

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

= \frac{3}{2} \cdot \int \frac{1}{x ( 1 + x )} d x

Apply u-substitution

: \int \frac{2}{1 + u ^{2}} d u

= \frac{3}{2} \cdot \int \frac{2}{1 + u ^{2}} d u

Take the constant out:

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

= \frac{3}{2} \cdot 2 \cdot \int \frac{1}{1 + u ^{2}} d u

Use the common integral:

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

= \frac{3}{2} \cdot 2 arctan (u)

Substitute back

u = x

= \frac{3}{2} \cdot 2 arctan (x)

Simplify

\frac{3}{2} \cdot 2 arctan (x) : 3 arctan (x)

= 3 arctan (x)

Add a constant to the solution

= 3 arctan (x) + C

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What is the integral of 3/(2sqrt(x)(1+x)) ?
The integral of 3/(2sqrt(x)(1+x)) is 3arctan(sqrt(x))+C