What is the integral of 5sec^2(5x)tan(5x) ?

The integral of 5sec^2(5x)tan(5x) is 1/2 tan^2(5x)+C

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integral of 5sec^2(5x)tan(5x)

\int 5 sec^{2} (5 x) tan (5 x) d x

\frac{1}{2} tan^{2} (5 x) + C

Solution steps

\int 5 sec^{2} (5 x) tan (5 x) d x

Take the constant out:

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

= 5 \cdot \int sec^{2} (5 x) tan (5 x) d x

Apply u-substitution

: \int \frac{u}{5} d u

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

Take the constant out:

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

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

Apply the Power Rule

: \frac{u ^{2}}{2}

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

Substitute back

u = tan (5 x)

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

Simplify

5 \cdot \frac{1}{5} \cdot \frac{tan ^{2} ( 5 x )}{2} : \frac{1}{2} tan^{2} (5 x)

= \frac{1}{2} tan^{2} (5 x)

Add a constant to the solution

= \frac{1}{2} tan^{2} (5 x) + C

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