What is the integral of (cos(x))/(sqrt(1+cos^2(x))) ?

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

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integral of (cos(x))/(sqrt(1+cos^2(x)))

\int \frac{cos ( x )}{1 + cos ^{2} ( x )} d x

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

Solution steps

\int \frac{cos ( x )}{1 + cos ^{2} ( x )} d x

Rewrite using trig identities

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

Apply u-substitution

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

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

Apply Trigonometric Substitution

: \int 1 d v

= \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} sin (x))

Simplify

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

Add a constant to the solution

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

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