What is the integral of sin(x)sqrt(1-4cos^2(x)) ?

The integral of sin(x)sqrt(1-4cos^2(x)) is -1/4 (arcsin(2cos(x))+1/2 sin(2arcsin(2cos(x))))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sin(x)sqrt(1-4cos^2(x))

\int sin (x) 1 - 4 cos^{2} (x) d x

- \frac{1}{4} (arcsin (2 cos (x)) + \frac{1}{2} sin (2 arcsin (2 cos (x)))) + C

Solution steps

\int sin (x) 1 - 4 cos^{2} (x) d x

Apply u-substitution

: \int - 1 - 4 u^{2} d u

= \int - 1 - 4 u^{2} d u

Take the constant out:

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

= - \int 1 - 4 u^{2} d u

Apply Trigonometric Substitution

: \int \frac{1}{2} cos^{2} (v) d v

= - \int \frac{1}{2} cos^{2} (v) d v

Take the constant out:

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

= - \frac{1}{2} \cdot \int cos^{2} (v) d v

Rewrite using trig identities

= - \frac{1}{2} \cdot \int \frac{1 + cos ( 2 v )}{2} d v

Take the constant out:

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

= - \frac{1}{2} \cdot \frac{1}{2} \cdot \int 1 + cos (2 v) d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \frac{1}{2} \cdot \frac{1}{2} (\int 1 d v + \int cos (2 v) d v)

\int 1 d v = v

\int cos (2 v) d v = \frac{1}{2} sin (2 v)

= - \frac{1}{2} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Substitute back

= - \frac{1}{2} \cdot \frac{1}{2} (arcsin (2 cos (x)) + \frac{1}{2} sin (2 arcsin (2 cos (x))))

Simplify

- \frac{1}{2} \cdot \frac{1}{2} (arcsin (2 cos (x)) + \frac{1}{2} sin (2 arcsin (2 cos (x)))) : - \frac{1}{4} (arcsin (2 cos (x)) + \frac{1}{2} sin (2 arcsin (2 cos (x))))

= - \frac{1}{4} (arcsin (2 cos (x)) + \frac{1}{2} sin (2 arcsin (2 cos (x))))

Add a constant to the solution

= - \frac{1}{4} (arcsin (2 cos (x)) + \frac{1}{2} sin (2 arcsin (2 cos (x)))) + C

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

\int sin^{4} (\frac{x}{3}) d x

\int \frac{5}{a + b x} d x

\int - 2 sin (x) + 5 cos (x) d x

\int e^{\frac{- x}{20}} d x

What is the integral of sin(x)sqrt(1-4cos^2(x)) ?
The integral of sin(x)sqrt(1-4cos^2(x)) is -1/4 (arcsin(2cos(x))+1/2 sin(2arcsin(2cos(x))))+C