What is the integral of (sin(θ)cos(θ))/(1+sin^2(θ)) ?

The integral of (sin(θ)cos(θ))/(1+sin^2(θ)) is 1/2 ln|1+sin^2(θ)|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (sin(θ)cos(θ))/(1+sin^2(θ))

\int \frac{sin ( θ ) cos ( θ )}{1 + sin ^{2} ( θ )} d θ

\frac{1}{2} ln 1 + sin^{2} (θ) + C

Solution steps

\int \frac{sin ( θ ) cos ( θ )}{1 + sin ^{2} ( θ )} d θ

Rewrite using trig identities

= \int \frac{sin ( 2 θ )}{2 ( 1 + sin ^{2} ( θ ) )} d θ

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 θ )}{1 + sin ^{2} ( θ )} d θ

Apply u-substitution

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

= \frac{1}{2} \cdot \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 = 1 + sin^{2} (θ)

= \frac{1}{2} ln 1 + sin^{2} (θ)

Add a constant to the solution

= \frac{1}{2} ln 1 + sin^{2} (θ) + C

\frac{\partial}{\partial x} (x^{3} + y^{3})

\frac{\partial}{\partial x} (ln (x^{2} + 3 y))

x \to \infty lim (1 - x^{2})

\frac{1}{2 - x} + \frac{1}{2 - 3 x}

\int \frac{7 x ^{2} - 6 x + 16}{x ^{3} + 4 x} d x

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