What is the integral of (sin(x))/(sqrt(4+cos^2(x))) ?

The integral of (sin(x))/(sqrt(4+cos^2(x))) is -ln(1/2 |cos(x)+sqrt(4+cos^2(x))|)+C

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integral of (sin(x))/(sqrt(4+cos^2(x)))

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

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

Solution steps

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

Apply u-substitution

: \int - \frac{1}{4 + u ^{2}} d u

= \int - \frac{1}{4 + 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}{4 + u ^{2}} d u

Apply Trigonometric Substitution

: \int sec (v) d v

= - \int sec (v) d v

Use the common integral:

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= - ln ∣ tan (v) + sec (v) ∣

Substitute back

= - ln tan (arctan (\frac{1}{2} cos (x))) + sec (arctan (\frac{1}{2} cos (x)))

Simplify

- ln tan (arctan (\frac{1}{2} cos (x))) + sec (arctan (\frac{1}{2} cos (x))) : - ln (\frac{1}{2} cos (x) + 4 + cos^{2} (x))

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

Add a constant to the solution

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

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What is the integral of (sin(x))/(sqrt(4+cos^2(x))) ?
The integral of (sin(x))/(sqrt(4+cos^2(x))) is -ln(1/2 |cos(x)+sqrt(4+cos^2(x))|)+C