What is the integral of (sin(cx))/(cos^2(cx)) ?

The integral of (sin(cx))/(cos^2(cx)) is 1/c sec(cx)+C

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integral of (sin(cx))/(cos^2(cx))

\int \frac{sin ( c x )}{cos ^{2} ( c x )} d x

\frac{1}{c} sec (c x) + C

Solution steps

\int \frac{sin ( c x )}{cos ^{2} ( c x )} d x

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

= \int tan (c x) sec (c x) d x

Apply u-substitution

= \int \frac{tan ( u ) sec ( u )}{c} d u

Take the constant out:

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

= \frac{1}{c} \cdot \int tan (u) sec (u) d u

Apply u-substitution

= \frac{1}{c} \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= \frac{1}{c} \cdot 1 \cdot v

Substitute back

= \frac{1}{c} \cdot 1 \cdot sec (c x)

Simplify

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

Add a constant to the solution

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

What is the integral of (sin(cx))/(cos^2(cx)) ?
The integral of (sin(cx))/(cos^2(cx)) is 1/c sec(cx)+C