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

The integral of (cos(x))/(a^2+sin^2(x)) is 1/a arctan((sin(x))/a)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

\frac{1}{a} arctan (\frac{sin ( x )}{a}) + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{a ^{2} + u ^{2}} d u

= \int \frac{1}{a ^{2} + u ^{2}} d u

Apply Integral Substitution

= \int \frac{1}{a ( v ^{2} + 1 )} d v

Take the constant out:

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

= \frac{1}{a} \cdot \int \frac{1}{v ^{2} + 1} d v

Use the common integral:

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= \frac{1}{a} arctan (v)

Substitute back

= \frac{1}{a} arctan (\frac{sin ( x )}{a})

Add a constant to the solution

= \frac{1}{a} arctan (\frac{sin ( x )}{a}) + C

\int \frac{1}{3 sin ( x )} d x

\int cos (x) 1 + sin (x) d x

\int e^{\frac{1}{x}} \frac{1}{x ^{2}} d x

\int \frac{1}{x + 8} d x

\int \frac{4}{cos ( x )} d x

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