What is the integral of arctan(a)x ?

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

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integral of arctan(a)x

\int arctan (a) x d x

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

Solution steps

\int arctan (a) x d x

Take the constant out:

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

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

Apply the Power Rule

: \frac{x ^{2}}{2}

= arctan (a) \frac{x ^{2}}{2}

Simplify

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

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

Add a constant to the solution

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

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