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

The integral of (cos(x))/(sqrt(4-cos^2(x))) is ln| 1/(sqrt(3))sin(x)+sqrt(1/3 (3+sin^2(x)))|+C

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

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

ln \frac{1}{3} sin (x) + \frac{1}{3} (3 + sin^{2} (x)) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{cos ( x )}{sin ^{2} ( x ) + 3} d x

Apply u-substitution

: \int \frac{1}{u ^{2} + 3} d u

= \int \frac{1}{u ^{2} + 3} 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}{3} sin (x))) + sec (arctan (\frac{1}{3} sin (x)))

Simplify

ln tan (arctan (\frac{1}{3} sin (x))) + sec (arctan (\frac{1}{3} sin (x))) : ln \frac{1}{3} sin (x) + \frac{1}{3} (3 + sin^{2} (x))

= ln \frac{1}{3} sin (x) + \frac{1}{3} (3 + sin^{2} (x))

Add a constant to the solution

= ln \frac{1}{3} sin (x) + \frac{1}{3} (3 + sin^{2} (x)) + C

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

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\int x a rc t a g (x) d x

\int (x + 4) 4 - x d x

\int (x^{3} + 2 x - 7) d x

What is the integral of (cos(x))/(sqrt(4-cos^2(x))) ?
The integral of (cos(x))/(sqrt(4-cos^2(x))) is ln| 1/(sqrt(3))sin(x)+sqrt(1/3 (3+sin^2(x)))|+C