What is the integral of sqrt(8+8cos(x)) ?

The integral of sqrt(8+8cos(x)) is 8tan(x/2)cos(x/2)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sqrt(8+8cos(x))

\int 8 + 8 cos (x) d x

8 tan (\frac{x}{2}) cos (\frac{x}{2}) + C

Solution steps

\int 8 + 8 cos (x) d x

8 + 8 cos (x) = 8 1 + cos (x)

= \int 8 1 + cos (x) d x

Take the constant out:

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

= 8 \cdot \int 1 + cos (x) d x

8 = 22

= 22 \cdot \int 1 + cos (x) d x

Apply u-substitution

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

= 22 \cdot \int \frac{2 2}{( 1 + u ^{2} ) 1 + u ^{2}} d u

Take the constant out:

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

= 22 \cdot 22 \cdot \int \frac{1}{( 1 + u ^{2} ) 1 + u ^{2}} d u

Simplify

(1 + u^{2}) 1 + u^{2} : (1 + u^{2})^{\frac{3}{2}}

= 22 \cdot 22 \cdot \int \frac{1}{( 1 + u ^{2} ) ^{\frac{3}{2}}} d u

Apply Trigonometric Substitution

: \int cos (v) d v

= 22 \cdot 22 \cdot \int cos (v) d v

Use the common integral:

\int cos (v) d v = sin (v)

= 22 \cdot 22 sin (v)

Substitute back

= 22 \cdot 22 sin (arctan (tan (\frac{x}{2})))

Simplify

22 \cdot 22 sin (arctan (tan (\frac{x}{2}))) : 8 tan (\frac{x}{2}) cos (\frac{x}{2})

= 8 tan (\frac{x}{2}) cos (\frac{x}{2})

Add a constant to the solution

= 8 tan (\frac{x}{2}) cos (\frac{x}{2}) + C

\int \frac{1}{x ^{3} 9 - x ^{2}} d x

\int (4 x^{3} - 5) d x

\int \frac{1}{x ( 1 - 7 x )} d x

\int \frac{3 t ^{2} - 3}{t ^{3} - 3 t} d t

\int e^{ux} d x

What is the integral of sqrt(8+8cos(x)) ?
The integral of sqrt(8+8cos(x)) is 8tan(x/2)cos(x/2)+C