What is the integral of sqrt(5+12x-9x^2) ?

The integral of sqrt(5+12x-9x^2) is 3/2 (arcsin(x-2/3)+1/2 sin(2arcsin(x-2/3)))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sqrt(5+12x-9x^2)

\int 5 + 12 x - 9 x^{2} d x

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

Solution steps

\int 5 + 12 x - 9 x^{2} d x

Complete the square

5 + 12 x - 9 x^{2} : - 9 (x - \frac{2}{3})^{2} + 9

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

Apply u-substitution

: \int 3 - u^{2} + 1 d u

= \int 3 - u^{2} + 1 d u

Take the constant out:

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

= 3 \cdot \int - u^{2} + 1 d u

Apply Trigonometric Substitution

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

= 3 \cdot \int cos^{2} (v) d v

Rewrite using trig identities

= 3 \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

= 3 \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

= 3 \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)

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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

\int 60 x d x

\int \frac{e ^{2 x}}{7 e ^{2 x} - 11} d x

\int x^{2} \cdot e^{x y} d y

\int 7 x arccos (x) d x

What is the integral of sqrt(5+12x-9x^2) ?
The integral of sqrt(5+12x-9x^2) is 3/2 (arcsin(x-2/3)+1/2 sin(2arcsin(x-2/3)))+C