What is the integral of (sin(x))/((cos(x))^4) ?

The integral of (sin(x))/((cos(x))^4) is 1/3 sec^3(x)+C

integral of (sin(x))/((cos(x))^4)

\int \frac{sin ( x )}{( cos ( x ) ) ^{4}} d x

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

Solution steps

\int \frac{sin ( x )}{( cos ( x ) ) ^{4}} d x

Simplify

= \int \frac{sin ( x )}{cos ^{4} ( x )} d x

Rewrite using trig identities

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

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

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

Apply u-substitution

= \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

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What is the integral of (sin(x))/((cos(x))^4) ?
The integral of (sin(x))/((cos(x))^4) is 1/3 sec^3(x)+C