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

The integral of (sin(x)cos(x))/(cos^2(x)) is -1/2 ln(cos^2(x))+C

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

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

- \frac{1}{2} ln (cos^{2} (x)) + C

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{sin ( 2 x )}{cos ^{2} ( x )} d x

Apply u-substitution

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

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

Take the constant out:

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

= \frac{1}{2} (- \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} (- ln ∣ u ∣)

Substitute back

u = cos^{2} (x)

= \frac{1}{2} (- ln cos^{2} (x))

Simplify

\frac{1}{2} (- ln cos^{2} (x)) : - \frac{1}{2} ln (cos^{2} (x))

= - \frac{1}{2} ln (cos^{2} (x))

Add a constant to the solution

= - \frac{1}{2} ln (cos^{2} (x)) + C

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What is the integral of (sin(x)cos(x))/(cos^2(x)) ?
The integral of (sin(x)cos(x))/(cos^2(x)) is -1/2 ln(cos^2(x))+C