What is the integral of cos(x)sqrt(25-sin^2(x)) ?

The integral of cos(x)sqrt(25-sin^2(x)) is 25/2 (arcsin(1/5 sin(x))+1/2 sin(2arcsin(1/5 sin(x))))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of cos(x)sqrt(25-sin^2(x))

\int cos (x) 25 - sin^{2} (x) d x

\frac{25}{2} (arcsin (\frac{1}{5} sin (x)) + \frac{1}{2} sin (2 arcsin (\frac{1}{5} sin (x)))) + C

Solution steps

\int cos (x) 25 - sin^{2} (x) d x

Apply u-substitution

= \int 25 - u^{2} d u

Apply Trigonometric Substitution

= \int 25 cos^{2} (v) d v

Take the constant out:

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

= 25 \cdot \int cos^{2} (v) d v

Rewrite using trig identities

= 25 \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

= 25 \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

= 25 \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)

= 25 \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Substitute back

= 25 \cdot \frac{1}{2} (arcsin (\frac{1}{5} sin (x)) + \frac{1}{2} sin (2 arcsin (\frac{1}{5} sin (x))))

Simplify

25 \cdot \frac{1}{2} (arcsin (\frac{1}{5} sin (x)) + \frac{1}{2} sin (2 arcsin (\frac{1}{5} sin (x)))) : \frac{25}{2} (arcsin (\frac{1}{5} sin (x)) + \frac{1}{2} sin (2 arcsin (\frac{1}{5} sin (x))))

= \frac{25}{2} (arcsin (\frac{1}{5} sin (x)) + \frac{1}{2} sin (2 arcsin (\frac{1}{5} sin (x))))

Add a constant to the solution

= \frac{25}{2} (arcsin (\frac{1}{5} sin (x)) + \frac{1}{2} sin (2 arcsin (\frac{1}{5} sin (x)))) + C

d er i v a t i v e f (t) = (9 t + 2)^{\frac{2}{3}}

d er i v a t i v e tan (6 x)

x \to 0 lim (\frac{sin ( 7 x )}{tan ( x )})

\int \frac{2 x ^{2} + x - 8}{x ^{3} - 4 x} d x

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

What is the integral of cos(x)sqrt(25-sin^2(x)) ?
The integral of cos(x)sqrt(25-sin^2(x)) is 25/2 (arcsin(1/5 sin(x))+1/2 sin(2arcsin(1/5 sin(x))))+C