What is the integral of ((arctan(7x))/(1+49x^2)) ?

The integral of ((arctan(7x))/(1+49x^2)) is 1/14 arctan^2(7x)+C

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integral of ((arctan(7x))/(1+49x^2))

\int (\frac{arctan ( 7 x )}{1 + 49 x ^{2}}) d x

\frac{1}{14} arctan^{2} (7 x) + C

Solution steps

\int \frac{arctan ( 7 x )}{1 + 49 x ^{2}} d x

Apply Integral Substitution

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

Take the constant out:

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

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

Apply u-substitution

: \int v d v

= \frac{1}{7} \cdot \int v d v

Apply the Power Rule

: \frac{v ^{2}}{2}

= \frac{1}{7} \cdot \frac{v ^{2}}{2}

Substitute back

= \frac{1}{7} \cdot \frac{arctan ^{2} ( 7 x )}{2}

Simplify

\frac{1}{7} \cdot \frac{arctan ^{2} ( 7 x )}{2} : \frac{1}{14} arctan^{2} (7 x)

= \frac{1}{14} arctan^{2} (7 x)

Add a constant to the solution

= \frac{1}{14} arctan^{2} (7 x) + C

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What is the integral of ((arctan(7x))/(1+49x^2)) ?
The integral of ((arctan(7x))/(1+49x^2)) is 1/14 arctan^2(7x)+C