What is the integral of 2tan(x)sec^3(x) ?

The integral of 2tan(x)sec^3(x) is 2/3 sec^3(x)+C

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integral of 2tan(x)sec^3(x)

\int 2 tan (x) sec^{3} (x) d x

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

Solution steps

\int 2 tan (x) sec^{3} (x) d x

Take the constant out:

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

= 2 \cdot \int tan (x) sec^{3} (x) d x

Apply u-substitution

= 2 \cdot \int \frac{1}{3} d u

Integral of a constant:

\int a d x = a x

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

Substitute back

u = sec^{3} (x)

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

Simplify

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

= \frac{2}{3} sec^{3} (x)

Add a constant to the solution

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

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