What is the integral of (arctan(x))/((1+x^2)^{3/2)} ?

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

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integral of (arctan(x))/((1+x^2)^{3/2)}

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

\frac{x arctan ( x )}{( 1 + x ^{2} ) ^{\frac{1}{2}}} + \frac{1}{( x ^{2} + 1 ) ^{\frac{1}{2}}} + C

Solution steps

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

Apply Integration By Parts

: \frac{x arctan ( x )}{( 1 + x ^{2} ) ^{\frac{1}{2}}} - \int \frac{x}{( x ^{2} + 1 ) ^{\frac{3}{2}}} d x

= \frac{x arctan ( x )}{( 1 + x ^{2} ) ^{\frac{1}{2}}} - \int \frac{x}{( x ^{2} + 1 ) ^{\frac{3}{2}}} d x

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

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

Simplify

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

Add a constant to the solution

= \frac{x arctan ( x )}{( 1 + x ^{2} ) ^{\frac{1}{2}}} + \frac{1}{( x ^{2} + 1 ) ^{\frac{1}{2}}} + C

\int 12 x^{2} (x^{3} + 4)^{3} d x

\frac{d}{d x} (x 2)

\frac{d}{d x} (x^{\frac{2}{3}} - 1)

x, x^{1}, 0, 1

n \to \infty lim (\frac{n}{4})

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