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

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

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

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

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

Solution steps

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

Apply u-substitution

= \int - \frac{1}{2 - u ^{2}} d u

Take the constant out:

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

= - \int \frac{1}{2 - u ^{2}} d u

Apply Trigonometric Substitution

= - \int 1 d v

Integral of a constant:

\int a d x = a x

= - 1 \cdot v

Substitute back

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

Simplify

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

Add a constant to the solution

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

d er i v a t i v e e^{4 x} cos (7 x)

\frac{d}{d x} (120 x)

x \to 0 lim (\frac{4 x ^{2} - 3 x}{x ^{2} + 0.8 x + \frac{4}{x}})

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

d er i v a t i v e (\frac{x + 8}{x - 8})^{5}

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