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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

u = cos^{2} (x)

= - ln cos^{2} (x)

Apply absolute rule:

x^{2} = x^{2}

, assuming

x \in R

cos^{2} (x) = cos^{2} (x)

= - ln (cos^{2} (x))

Add a constant to the solution

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

x \to 0 lim (sin^{2} (x))

e^{3 t s i n (2 t)}

x \to 7 + lim (ln (x^{2} - 49))

\int 4 x^{9} d x

y = sin^{2} (x), y = sin^{3} (x), x = 0, x = π

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