What is the integral of sqrt(1-2x-x^2) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sqrt(1-2x-x^2)

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

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

Solution steps

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

Complete the square

1 - 2 x - x^{2} : - (x + 1)^{2} + 2

= \int - (x + 1)^{2} + 2 d x

Apply u-substitution

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

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

Apply Trigonometric Substitution

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

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

Take the constant out:

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

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

Rewrite using trig identities

= 2 \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

= 2 \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

= 2 \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)

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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

\int \frac{1}{( 11 + 5 x ) ^{3}} d x

\int \frac{x ^{4}}{( x ^{2} + 1 )} d x

\int (x + 4) (x - 3) d x

\int (x (2 x + 5)^{8}) d x

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