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

The integral of sqrt(tan(4x))sec^2(4x) is 1/6 tan^{3/2}(4x)+C

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

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

\frac{1}{6} tan^{\frac{3}{2}} (4 x) + C

Solution steps

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

Apply u-substitution

= \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

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

Apply the Power Rule

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

Substitute back

u = tan (4 x)

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

Simplify

\frac{1}{2} \cdot \frac{( tan ( 4 x ) ) ^{3}}{3} : \frac{1}{6} tan^{\frac{3}{2}} (4 x)

= \frac{1}{6} tan^{\frac{3}{2}} (4 x)

Add a constant to the solution

= \frac{1}{6} tan^{\frac{3}{2}} (4 x) + C

\int 15 x \cdot 1 - x d x

d er i v a t i v e \frac{4 - sec ( x )}{tan ( x )}

x \to 0 lim (\frac{1}{ln ( 1 + x )} - \frac{1}{x})

d er i v a t i v e (x^{4} + 3 x)^{- 1}

\frac{d}{d x} (x^{2} + \frac{3456}{x})

What is the integral of sqrt(tan(4x))sec^2(4x) ?
The integral of sqrt(tan(4x))sec^2(4x) is 1/6 tan^{3/2}(4x)+C