What is the integral of sin(arccos(x)) ?

The integral of sin(arccos(x)) is 1/2 (arcsin(x)+1/2 sin(2arcsin(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(arccos(x))

\int sin (arccos (x)) d x

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

Solution steps

\int sin (arccos (x)) d x

Use the following identity:

sin (arccos (x)) = 1 - x^{2}

= \int 1 - x^{2} d x

Apply Trigonometric Substitution

: \int cos^{2} (u) d u

= \int cos^{2} (u) d u

Rewrite using trig identities

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

Take the constant out:

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

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

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} (\int 1 d u + \int cos (2 u) d u)

\int 1 d u = u

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

= \frac{1}{2} (u + \frac{1}{2} sin (2 u))

Substitute back

u = arcsin (x)

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

Add a constant to the solution

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

\int 3 e^{- t} d t

5 x y \frac{d y}{d x} = 4

\frac{e ^{- s}}{s ^{2}}

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

\frac{\partial}{\partial x} (\frac{e ^{7 - x}}{y ^{5}})

What is the integral of sin(arccos(x)) ?
The integral of sin(arccos(x)) is 1/2 (arcsin(x)+1/2 sin(2arcsin(x)))+C