What is the integral of (sin(x))/(5+cos^2(x)) ?

The integral of (sin(x))/(5+cos^2(x)) is -1/(sqrt(5))arctan((cos(x))/(sqrt(5)))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (sin(x))/(5+cos^2(x))

\int \frac{sin ( x )}{5 + cos ^{2} ( x )} d x

- \frac{1}{5} arctan (\frac{cos ( x )}{5}) + C

Solution steps

\int \frac{sin ( x )}{5 + cos ^{2} ( x )} d x

Apply u-substitution

: \int - \frac{1}{5 + u ^{2}} d u

= \int - \frac{1}{5 + u ^{2}} d u

Take the constant out:

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

= - \int \frac{1}{5 + u ^{2}} d u

Apply Integral Substitution

= - \int \frac{1}{5 ( 1 + v ^{2} )} d v

Take the constant out:

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

= - \frac{1}{5} \cdot \int \frac{1}{1 + v ^{2}} d v

Use the common integral:

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

= - \frac{1}{5} arctan (v)

Substitute back

= - \frac{1}{5} arctan (\frac{cos ( x )}{5})

Add a constant to the solution

= - \frac{1}{5} arctan (\frac{cos ( x )}{5}) + C

\frac{d}{d x} (1 - ∣ 1 - x ∣)

\int \frac{x ^{4}}{x} d x

(x^{2} + 2)^{2} y = \frac{5 x ( y ^{2} + 1 ) ^{2}}{( x ^{2} + 2 ) y ^{^{'}}} - (x^{4} + 4 x^{2} + 4) y

\frac{d y}{d x} = \frac{1}{x ^{4} + 4 x ^{2}}

x \to 2 lim (sin (\frac{1}{x - 2}))

What is the integral of (sin(x))/(5+cos^2(x)) ?
The integral of (sin(x))/(5+cos^2(x)) is -1/(sqrt(5))arctan((cos(x))/(sqrt(5)))+C