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

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

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

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

\frac{2}{5} arctan (\frac{1}{5} sin (x)) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

Apply Integral Substitution

= 2 \cdot \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

= 2 \cdot \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)

= 2 \cdot \frac{1}{5} arctan (v)

Substitute back

= 2 \cdot \frac{1}{5} arctan (\frac{sin ( x )}{5})

Simplify

2 \cdot \frac{1}{5} arctan (\frac{sin ( x )}{5}) : \frac{2}{5} arctan (\frac{1}{5} sin (x))

= \frac{2}{5} arctan (\frac{1}{5} sin (x))

Add a constant to the solution

= \frac{2}{5} arctan (\frac{1}{5} sin (x)) + C

\int 70 x e^{7 x^{2}} d x

x \to 0 lim (\frac{2 - x}{x})

sec (3 x)

\frac{x ^{2} + 1}{x + 4}

\frac{1}{6 s}

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