What is the integral of (4sinh(x))/(4+cosh^2(x)) ?

The integral of (4sinh(x))/(4+cosh^2(x)) is 2arctan(1/2 cosh(x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (4sinh(x))/(4+cosh^2(x))

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

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

Solution steps

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

Take the constant out:

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

= 4 \cdot \int \frac{sinh ( x )}{4 + cosh ^{2} ( x )} d x

Apply u-substitution

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

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

Apply Integral Substitution

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

Take the constant out:

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

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

Use the common integral:

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

= 4 \cdot \frac{1}{2} arctan (v)

Substitute back

= 4 \cdot \frac{1}{2} arctan (\frac{cosh ( x )}{2})

Simplify

4 \cdot \frac{1}{2} arctan (\frac{cosh ( x )}{2}) : 2 arctan (\frac{1}{2} cosh (x))

= 2 arctan (\frac{1}{2} cosh (x))

Add a constant to the solution

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

\frac{d y}{d x}, y = 3 x

\frac{d}{d x} (x + \frac{9}{x})

\int (3 - 8 x) d x

\frac{d}{d x} (\frac{2 x ^{3} + 8}{x})

What is the integral of (4sinh(x))/(4+cosh^2(x)) ?
The integral of (4sinh(x))/(4+cosh^2(x)) is 2arctan(1/2 cosh(x))+C