What is the integral of (x^2)/(sqrt(289-x^2)) ?

The integral of (x^2)/(sqrt(289-x^2)) is 289/2 (arcsin(1/17 x)-1/2 sin(2arcsin(1/17 x)))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^2)/(sqrt(289-x^2))

\int \frac{x ^{2}}{289 - x ^{2}} d x

\frac{289}{2} (arcsin (\frac{1}{17} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{17} x))) + C

Solution steps

\int \frac{x ^{2}}{289 - x ^{2}} d x

Apply Trigonometric Substitution

: \int 289 sin^{2} (u) d u

= \int 289 sin^{2} (u) d u

Take the constant out:

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

= 289 \cdot \int sin^{2} (u) d u

Rewrite using trig identities

= 289 \cdot \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

= 289 \cdot \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

= 289 \cdot \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)

= 289 \cdot \frac{1}{2} (u - \frac{1}{2} sin (2 u))

Substitute back

u = arcsin (\frac{1}{17} x)

= 289 \cdot \frac{1}{2} (arcsin (\frac{1}{17} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{17} x)))

Simplify

289 \cdot \frac{1}{2} (arcsin (\frac{1}{17} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{17} x))) : \frac{289}{2} (arcsin (\frac{1}{17} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{17} x)))

= \frac{289}{2} (arcsin (\frac{1}{17} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{17} x)))

Add a constant to the solution

= \frac{289}{2} (arcsin (\frac{1}{17} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{17} x))) + C

\frac{d ^{2}}{d y ^{2}} (x y e^{(- x + 3 y)})

(0.4) (2)

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

\frac{\partial}{\partial x} (7 tan (x y))

n \to \infty lim (1)

What is the integral of (x^2)/(sqrt(289-x^2)) ?
The integral of (x^2)/(sqrt(289-x^2)) is 289/2 (arcsin(1/17 x)-1/2 sin(2arcsin(1/17 x)))+C