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

The integral of (x^2a)/((1+a^2x^2)) is a(-1/(a^3)arctan(ax)+x/(a^2))+C

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

integral

\frac{x ^{2} a}{( 1 + a ^{2} x ^{2} )}

a (- \frac{1}{a ^{3}} arctan (a x) + \frac{x}{a ^{2}}) + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Expand

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

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

Add a constant to the solution

= a (- \frac{1}{a ^{3}} arctan (a x) + \frac{x}{a ^{2}}) + C

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

1 + e^{x}

\frac{cos ^{2} ( x )}{( 1 + sin ^{2} ( x ) )}

\frac{x}{( x ^{4} - 1 )}

\frac{200}{x ^{3}}

What is the integral of (x^2a)/((1+a^2x^2)) ?
The integral of (x^2a)/((1+a^2x^2)) is a(-1/(a^3)arctan(ax)+x/(a^2))+C