What is the integral of (sin(x))/(sqrt(7-cos^2(x))) ?

The integral of (sin(x))/(sqrt(7-cos^2(x))) is -arcsin(1/(sqrt(7))cos(x))+C

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integral of (sin(x))/(sqrt(7-cos^2(x)))

\int \frac{sin ( x )}{7 - cos ^{2} ( x )} d x

- arcsin (\frac{1}{7} cos (x)) + C

Solution steps

\int \frac{sin ( x )}{7 - cos ^{2} ( x )} d x

Apply u-substitution

= \int - \frac{1}{7 - 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}{7 - 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}{7} cos (x))

Simplify

= - arcsin (\frac{1}{7} cos (x))

Add a constant to the solution

= - arcsin (\frac{1}{7} cos (x)) + C

What is the integral of (sin(x))/(sqrt(7-cos^2(x))) ?
The integral of (sin(x))/(sqrt(7-cos^2(x))) is -arcsin(1/(sqrt(7))cos(x))+C