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

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

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integral of (cos(5x))/(1+sin^2(5x))

\int \frac{cos ( 5 x )}{1 + sin ^{2} ( 5 x )} d x

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

Solution steps

\int \frac{cos ( 5 x )}{1 + sin ^{2} ( 5 x )} d x

Apply u-substitution

: \int \frac{cos ( u )}{5 ( 1 + sin ^{2} ( u ) )} d u

= \int \frac{cos ( u )}{5 ( 1 + sin ^{2} ( u ) )} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{cos ( u )}{1 + sin ^{2} ( u )} d u

Apply u-substitution

: \int \frac{1}{1 + v ^{2}} d v

= \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 (sin (5 x))

Add a constant to the solution

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

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\int \frac{x ^{2} - 1}{x ^{2} + 1} d x

y^{^{''}} - 16 y^{^{'}} + 100 y = 0

\frac{\partial}{\partial y} (4 x y^{3} (z^{2} + 1)^{3})

f (x) = \frac{e ^{3 x}}{x ^{15}}

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