What is the integral of (x^2)/(sqrt(361-x^2)) ?

The integral of (x^2)/(sqrt(361-x^2)) is 361/2 (arcsin(1/19 x)-1/2 sin(2arcsin(1/19 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(361-x^2))

\int \frac{x ^{2}}{361 - x ^{2}} d x

\frac{361}{2} (arcsin (\frac{1}{19} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{19} x))) + C

Solution steps

\int \frac{x ^{2}}{361 - x ^{2}} d x

Apply Trigonometric Substitution

: \int 361 sin^{2} (u) d u

= \int 361 sin^{2} (u) d u

Take the constant out:

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

= 361 \cdot \int sin^{2} (u) d u

Rewrite using trig identities

= 361 \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

= 361 \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

= 361 \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)

= 361 \cdot \frac{1}{2} (u - \frac{1}{2} sin (2 u))

Substitute back

u = arcsin (\frac{1}{19} x)

= 361 \cdot \frac{1}{2} (arcsin (\frac{1}{19} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{19} x)))

Simplify

361 \cdot \frac{1}{2} (arcsin (\frac{1}{19} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{19} x))) : \frac{361}{2} (arcsin (\frac{1}{19} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{19} x)))

= \frac{361}{2} (arcsin (\frac{1}{19} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{19} x)))

Add a constant to the solution

= \frac{361}{2} (arcsin (\frac{1}{19} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{19} x))) + C

y^{^{''}} - 6 y^{^{'}} + 9 y = t^{- 9} e^{3 t}

\int \frac{4 x + 2}{x ^{2} + x + 5} d x

5 x - x^{2} (1.4)

\frac{\partial}{\partial x} (x^{2} - 4 y^{2})

x \to + 1 lim (x^{3} - 1)

What is the integral of (x^2)/(sqrt(361-x^2)) ?
The integral of (x^2)/(sqrt(361-x^2)) is 361/2 (arcsin(1/19 x)-1/2 sin(2arcsin(1/19 x)))+C