What is the integral of sqrt(1+cos(θ)) ?

The integral of sqrt(1+cos(θ)) is 2sqrt(2)tan(θ/2)cos(θ/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(1+cos(θ))

\int 1 + cos (θ) d θ

22 tan (\frac{θ}{2}) cos (\frac{θ}{2}) + C

Solution steps

\int 1 + cos (θ) d θ

Apply u-substitution

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

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

Apply Trigonometric Substitution

: \int cos (v) d v

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

Use the common integral:

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

= 22 sin (v)

Substitute back

= 22 sin (arctan (tan (\frac{θ}{2})))

Simplify

22 sin (arctan (tan (\frac{θ}{2}))) : 22 tan (\frac{θ}{2}) cos (\frac{θ}{2})

= 22 tan (\frac{θ}{2}) cos (\frac{θ}{2})

Add a constant to the solution

= 22 tan (\frac{θ}{2}) cos (\frac{θ}{2}) + C

\int \frac{1}{y 1 - y ^{2}} d y

\int 3 4 - x^{2} d x

\int sec^{2} (6 x) tan^{2} (6 x) d x

\int 8 t e^{2 t} d t

\int \frac{1 + ln ( x )}{3 + x ln ( x )} d x

What is the integral of sqrt(1+cos(θ)) ?
The integral of sqrt(1+cos(θ)) is 2sqrt(2)tan(θ/2)cos(θ/2)+C