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

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

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

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

22 (- \frac{1}{8} (ln 2 x + 22 x + 2 - 2 arctan (2 x + 1)) + \frac{1}{8} (ln 2 x - 22 x + 2 + 2 arctan (2 x - 1))) + C

Solution steps

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

Apply Integral Substitution

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{2 v ^{2}}{v ^{4} + 4} d v

= 2 \cdot \int \frac{2 v ^{2}}{v ^{4} + 4} d v

Take the constant out:

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

= 2 \cdot 2 \cdot \int \frac{v ^{2}}{v ^{4} + 4} d v

Take the partial fraction of

\frac{v ^{2}}{v ^{4} + 4} : - \frac{v}{4 ( v ^{2} + 2 v + 2 )} + \frac{v}{4 ( v ^{2} - 2 v + 2 )}

= 2 \cdot 2 \cdot \int - \frac{v}{4 ( v ^{2} + 2 v + 2 )} + \frac{v}{4 ( v ^{2} - 2 v + 2 )} d v

Apply the Sum Rule:

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

= 2 \cdot 2 (- \int \frac{v}{4 ( v ^{2} + 2 v + 2 )} d v + \int \frac{v}{4 ( v ^{2} - 2 v + 2 )} d v)

\int \frac{v}{4 ( v ^{2} + 2 v + 2 )} d v = \frac{1}{8} (ln v^{2} + 2 v + 2 - 2 arctan (v + 1))

\int \frac{v}{4 ( v ^{2} - 2 v + 2 )} d v = \frac{1}{8} (ln v^{2} - 2 v + 2 + 2 arctan (v - 1))

= 2 \cdot 2 (- \frac{1}{8} (ln v^{2} + 2 v + 2 - 2 arctan (v + 1)) + \frac{1}{8} (ln v^{2} - 2 v + 2 + 2 arctan (v - 1)))

Substitute back

= 2 \cdot 2 (- \frac{1}{8} (ln (2 x)^{2} + 2 2 x + 2 - 2 arctan (2 x + 1)) + \frac{1}{8} (ln (2 x)^{2} - 2 2 x + 2 + 2 arctan (2 x - 1)))

Simplify

2 \cdot 2 (- \frac{1}{8} (ln (2 x)^{2} + 2 2 x + 2 - 2 arctan (2 x + 1)) + \frac{1}{8} (ln (2 x)^{2} - 2 2 x + 2 + 2 arctan (2 x - 1))) : 22 (- \frac{1}{8} (ln 2 x + 22 x + 2 - 2 arctan (2 x + 1)) + \frac{1}{8} (ln 2 x - 22 x + 2 + 2 arctan (2 x - 1)))

= 22 (- \frac{1}{8} (ln 2 x + 22 x + 2 - 2 arctan (2 x + 1)) + \frac{1}{8} (ln 2 x - 22 x + 2 + 2 arctan (2 x - 1)))

Add a constant to the solution

= 22 (- \frac{1}{8} (ln 2 x + 22 x + 2 - 2 arctan (2 x + 1)) + \frac{1}{8} (ln 2 x - 22 x + 2 + 2 arctan (2 x - 1))) + C

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