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

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

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

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

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

Solution steps

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

Apply u-substitution

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

= \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{sin ( x )}{5})

Add a constant to the solution

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

\frac{d y}{d x} = 6 e^{x - y}

\frac{d}{d x} (x^{3} + x)

n = 1 \sum \infty \frac{1}{n ^{2} + 5}

\int \frac{1}{( 4 + x ^{2} )} d x

y = \frac{2}{3} (x^{2} + 1)^{\frac{3}{2}}

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