What is the integral of (sin(5x))/(cos^4(5x)) ?

The integral of (sin(5x))/(cos^4(5x)) is 1/15 sec^3(5x)+C

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integral of (sin(5x))/(cos^4(5x))

\int \frac{sin ( 5 x )}{cos ^{4} ( 5 x )} d x

\frac{1}{15} sec^{3} (5 x) + C

Solution steps

\int \frac{sin ( 5 x )}{cos ^{4} ( 5 x )} d x

Rewrite using trig identities

= \int \frac{tan ( 5 x )}{cos ^{3} ( 5 x )} d x

Use the basic trigonometric identity:

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

= \int sec^{3} (5 x) tan (5 x) d x

Apply u-substitution

: \int sec^{3} (u) tan (u) \frac{1}{5} d u

= \int sec^{3} (u) tan (u) \frac{1}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int sec^{3} (u) tan (u) d u

Apply u-substitution

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

= \frac{1}{5} \cdot \int \frac{1}{3} d v

Integral of a constant:

\int a d x = a x

= \frac{1}{5} \cdot \frac{1}{3} v

Substitute back

= \frac{1}{5} \cdot \frac{1}{3} sec^{3} (5 x)

Simplify

\frac{1}{5} \cdot \frac{1}{3} sec^{3} (5 x) : \frac{1}{15} sec^{3} (5 x)

= \frac{1}{15} sec^{3} (5 x)

Add a constant to the solution

= \frac{1}{15} sec^{3} (5 x) + C

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What is the integral of (sin(5x))/(cos^4(5x)) ?
The integral of (sin(5x))/(cos^4(5x)) is 1/15 sec^3(5x)+C