What is the integral of (4-x)/(sqrt(4-x^2)) ?

The integral of (4-x)/(sqrt(4-x^2)) is sqrt(4-x^2)+4arcsin(1/2 x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (4-x)/(sqrt(4-x^2))

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

4 - x^{2} + 4 arcsin (\frac{1}{2} x) + C

Solution steps

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

Apply Trigonometric Substitution

: \int 2 (- sin (u) + 2) d u

= \int 2 (- sin (u) + 2) d u

Take the constant out:

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

= 2 \cdot \int - sin (u) + 2 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

= 2 (- \int sin (u) d u + \int 2 d u)

\int sin (u) d u = - cos (u)

\int 2 d u = 2 u

= 2 (- (- cos (u)) + 2 u)

Substitute back

u = arcsin (\frac{1}{2} x)

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

Simplify

2 (- (- cos (arcsin (\frac{1}{2} x))) + 2 arcsin (\frac{1}{2} x)) : 4 - x^{2} + 4 arcsin (\frac{1}{2} x)

= 4 - x^{2} + 4 arcsin (\frac{1}{2} x)

Add a constant to the solution

= 4 - x^{2} + 4 arcsin (\frac{1}{2} x) + C

\int 30 csc (5 x) sec (5 x) d x

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

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

\int \frac{- 7 x ^{2} - 9}{x ^{3}} d x

\int \frac{x ^{3}}{( 2 - x ^{4} ) ^{2}} d x

What is the integral of (4-x)/(sqrt(4-x^2)) ?
The integral of (4-x)/(sqrt(4-x^2)) is sqrt(4-x^2)+4arcsin(1/2 x)+C