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

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

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

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

- arctan (x) + x + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

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

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

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

\int 1 d u = u

= \frac{1}{2} \cdot 2 (- arctan (u) + u)

Substitute back

u = x

= \frac{1}{2} \cdot 2 (- arctan (x) + x)

Simplify

\frac{1}{2} \cdot 2 (- arctan (x) + x) : - arctan (x) + x

= - arctan (x) + x

Add a constant to the solution

= - arctan (x) + x + C

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