What is the integral of 4sec^2(4x)sqrt(tan(4x)) ?

The integral of 4sec^2(4x)sqrt(tan(4x)) is 2/3 tan^{3/2}(4x)+C

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integral of 4sec^2(4x)sqrt(tan(4x))

\int 4 sec^{2} (4 x) tan (4 x) d x

\frac{2}{3} tan^{\frac{3}{2}} (4 x) + C

Solution steps

\int 4 sec^{2} (4 x) tan (4 x) d x

Take the constant out:

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

= 4 \cdot \int sec^{2} (4 x) tan (4 x) d x

Apply u-substitution

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

= 4 \cdot \int \frac{u ^{2}}{2} d u

Take the constant out:

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

= 4 \cdot \frac{1}{2} \cdot \int u^{2} d u

Apply the Power Rule

: \frac{u ^{3}}{3}

= 4 \cdot \frac{1}{2} \cdot \frac{u ^{3}}{3}

Substitute back

u = tan (4 x)

= 4 \cdot \frac{1}{2} \cdot \frac{( tan ( 4 x ) ) ^{3}}{3}

Simplify

4 \cdot \frac{1}{2} \cdot \frac{( tan ( 4 x ) ) ^{3}}{3} : \frac{2}{3} tan^{\frac{3}{2}} (4 x)

= \frac{2}{3} tan^{\frac{3}{2}} (4 x)

Add a constant to the solution

= \frac{2}{3} tan^{\frac{3}{2}} (4 x) + C

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What is the integral of 4sec^2(4x)sqrt(tan(4x)) ?
The integral of 4sec^2(4x)sqrt(tan(4x)) is 2/3 tan^{3/2}(4x)+C