What is the integral of ((1-((x^2)/9)))^{1/2} ?

The integral of ((1-((x^2)/9)))^{1/2} is 3/2 (arcsin(1/3 x)+1/2 sin(2arcsin(1/3 x)))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of ((1-((x^2)/9)))^{1/2}

\int ((1 - (\frac{x ^{2}}{9})))^{\frac{1}{2}} d x

\frac{3}{2} (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) + C

Solution steps

\int (1 - \frac{x ^{2}}{9})^{\frac{1}{2}} d x

Simplify

(1 - \frac{x ^{2}}{9})^{\frac{1}{2}} : \frac{1}{3} (9 - x^{2})^{\frac{1}{2}}

= \int \frac{1}{3} (9 - x^{2})^{\frac{1}{2}} d x

Take the constant out:

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

= \frac{1}{3} \cdot \int (9 - x^{2})^{\frac{1}{2}} d x

Apply Trigonometric Substitution

: \int 9 cos^{2} (u) d u

= \frac{1}{3} \cdot \int 9 cos^{2} (u) d u

Take the constant out:

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

= \frac{1}{3} \cdot 9 \cdot \int cos^{2} (u) d u

Rewrite using trig identities

= \frac{1}{3} \cdot 9 \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

= \frac{1}{3} \cdot 9 \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

= \frac{1}{3} \cdot 9 \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)

= \frac{1}{3} \cdot 9 \cdot \frac{1}{2} (u + \frac{1}{2} sin (2 u))

Substitute back

u = arcsin (\frac{1}{3} x)

= \frac{1}{3} \cdot 9 \cdot \frac{1}{2} (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x)))

Simplify

\frac{1}{3} \cdot 9 \cdot \frac{1}{2} (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) : \frac{3}{2} (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x)))

= \frac{3}{2} (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x)))

Add a constant to the solution

= \frac{3}{2} (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) + C

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

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

\frac{d}{d x} (e^{\frac{x ^{4}}{10}})

\frac{d y}{d x} = x^{2} + y - 1

y^{^{''}} + 9 y = 2 sin (2 x), y (0) = 0, y^{^{'}} (0) = - 1

What is the integral of ((1-((x^2)/9)))^{1/2} ?
The integral of ((1-((x^2)/9)))^{1/2} is 3/2 (arcsin(1/3 x)+1/2 sin(2arcsin(1/3 x)))+C