What is the integral of ((arctan(x))^7)/(1+x^2) ?

The integral of ((arctan(x))^7)/(1+x^2) is (arctan^8(x))/8+C

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integral of ((arctan(x))^7)/(1+x^2)

\int \frac{( arctan ( x ) ) ^{7}}{1 + x ^{2}} d x

\frac{arctan ^{8} ( x )}{8} + C

Solution steps

\int \frac{( arctan ( x ) ) ^{7}}{1 + x ^{2}} d x

Simplify

= \int \frac{arctan ^{7} ( x )}{1 + x ^{2}} d x

Apply u-substitution

: \int u^{7} d u

= \int u^{7} d u

Apply the Power Rule

: \frac{u ^{8}}{8}

= \frac{u ^{8}}{8}

Substitute back

u = arctan (x)

= \frac{arctan ^{8} ( x )}{8}

Add a constant to the solution

= \frac{arctan ^{8} ( x )}{8} + C

x \to 1 lim (\frac{1 - x}{x - 1})

f (x) = 5 x^{4} e^{x} + e^{x} x^{5}

4 + x^{2} + 4 x^{4}, - 3

f (t) = sin (4 t)

\frac{d}{d x} ((3 x)^{6 x})

What is the integral of ((arctan(x))^7)/(1+x^2) ?
The integral of ((arctan(x))^7)/(1+x^2) is (arctan^8(x))/8+C