What is the integral of (sin(26x))/(1+cos^2(26x)) ?

The integral of (sin(26x))/(1+cos^2(26x)) is 1/26 arctan(tan^2(13x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (sin(26x))/(1+cos^2(26x))

\int \frac{sin ( 26 x )}{1 + cos ^{2} ( 26 x )} d x

\frac{1}{26} arctan (tan^{2} (13 x)) + C

Solution steps

\int \frac{sin ( 26 x )}{1 + cos ^{2} ( 26 x )} d x

Apply u-substitution

: \int \frac{sin ( u )}{26 ( 1 + cos ^{2} ( u ) )} d u

= \int \frac{sin ( u )}{26 ( 1 + cos ^{2} ( u ) )} d u

Take the constant out:

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

= \frac{1}{26} \cdot \int \frac{sin ( u )}{1 + cos ^{2} ( u )} d u

Rewrite using trig identities

= \frac{1}{26} \cdot \int \frac{sin ( u )}{- sin ^{2} ( u ) + 2} d u

Apply u-substitution

: \int \frac{2 v}{v ^{4} + 1} d v

= \frac{1}{26} \cdot \int \frac{2 v}{v ^{4} + 1} d v

Take the constant out:

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

= \frac{1}{26} \cdot 2 \cdot \int \frac{v}{v ^{4} + 1} d v

Apply u-substitution

: \int \frac{1}{2 ( w ^{2} + 1 )} d w

= \frac{1}{26} \cdot 2 \cdot \int \frac{1}{2 ( w ^{2} + 1 )} d w

Take the constant out:

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

= \frac{1}{26} \cdot 2 \cdot \frac{1}{2} \cdot \int \frac{1}{w ^{2} + 1} d w

Use the common integral:

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

= \frac{1}{26} \cdot 2 \cdot \frac{1}{2} arctan (w)

Substitute back

= \frac{1}{26} \cdot 2 \cdot \frac{1}{2} arctan (tan^{2} (\frac{26 x}{2}))

Simplify

\frac{1}{26} \cdot 2 \cdot \frac{1}{2} arctan (tan^{2} (\frac{26 x}{2})) : \frac{1}{26} arctan (tan^{2} (13 x))

= \frac{1}{26} arctan (tan^{2} (13 x))

Add a constant to the solution

= \frac{1}{26} arctan (tan^{2} (13 x)) + C

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

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

\int 3 \cdot e^{x} \cdot (x^{2} + 4) d x

\int \frac{2 lo g _{4} ( x )}{x} d x

\int \frac{1}{x - 100} d x

What is the integral of (sin(26x))/(1+cos^2(26x)) ?
The integral of (sin(26x))/(1+cos^2(26x)) is 1/26 arctan(tan^2(13x))+C