What is the integral of (-5arctan(x))/(2+2x^2) ?

The integral of (-5arctan(x))/(2+2x^2) is -5/4 arctan^2(x)+C

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integral of (-5arctan(x))/(2+2x^2)

\int \frac{- 5 arctan ( x )}{2 + 2 x ^{2}} d x

- \frac{5}{4} arctan^{2} (x) + C

Solution steps

\int \frac{- 5 arctan ( x )}{2 + 2 x ^{2}} d x

Take the constant out:

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

= - 5 \cdot \int \frac{arctan ( x )}{2 + 2 x ^{2}} d x

Apply u-substitution

= - 5 \cdot \int \frac{u}{2} d u

Take the constant out:

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

= - 5 \cdot \frac{1}{2} \cdot \int u d u

Apply the Power Rule

= - 5 \cdot \frac{1}{2} \cdot \frac{u ^{2}}{2}

Substitute back

u = arctan (x)

= - 5 \cdot \frac{1}{2} \cdot \frac{arctan ^{2} ( x )}{2}

Simplify

- 5 \cdot \frac{1}{2} \cdot \frac{arctan ^{2} ( x )}{2} : - \frac{5}{4} arctan^{2} (x)

= - \frac{5}{4} arctan^{2} (x)

Add a constant to the solution

= - \frac{5}{4} arctan^{2} (x) + C

What is the integral of (-5arctan(x))/(2+2x^2) ?
The integral of (-5arctan(x))/(2+2x^2) is -5/4 arctan^2(x)+C