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

The integral of (sec(x))^3tan(x) is 1/3 sec^3(x)+C

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

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

\frac{1}{3} sec^{3} (x) + C

Solution steps

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

Simplify

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

Apply u-substitution

: \int \frac{1}{3} d u

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

Integral of a constant:

\int a d x = a x

= \frac{1}{3} u

Substitute back

u = sec^{3} (x)

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

Add a constant to the solution

= \frac{1}{3} sec^{3} (x) + C

2 y^{3}

\int (x^{3} - a^{3})^{5} d x

\int - 8 cos (t) d t

\int 3 x^{3} - x d x

\int (\frac{x}{3} + \frac{8}{x}) d x

What is the integral of (sec(x))^3tan(x) ?
The integral of (sec(x))^3tan(x) is 1/3 sec^3(x)+C