What is the integral of ((arctan(x))^5)/(1+x^2) ?

The integral of ((arctan(x))^5)/(1+x^2) is (arctan^6(x))/6+C

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integral of ((arctan(x))^5)/(1+x^2)

\int \frac{( arctan ( x ) ) ^{5}}{1 + x ^{2}} d x

\frac{arctan ^{6} ( x )}{6} + C

Solution steps

\int \frac{( arctan ( x ) ) ^{5}}{1 + x ^{2}} d x

Apply u-substitution

: \int u^{5} d u

= \int u^{5} d u

Apply the Power Rule

: \frac{u ^{6}}{6}

= \frac{u ^{6}}{6}

Substitute back

u = arctan (x)

= \frac{arctan ^{6} ( x )}{6}

Add a constant to the solution

= \frac{arctan ^{6} ( x )}{6} + C

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What is the integral of ((arctan(x))^5)/(1+x^2) ?
The integral of ((arctan(x))^5)/(1+x^2) is (arctan^6(x))/6+C