What is the integral of (cos(8x))/(1+sin^2(8x)) ?

The integral of (cos(8x))/(1+sin^2(8x)) is 1/8 arctan(sin(8x))+C

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integral of (cos(8x))/(1+sin^2(8x))

\int \frac{cos ( 8 x )}{1 + sin ^{2} ( 8 x )} d x

\frac{1}{8} arctan (sin (8 x)) + C

Solution steps

\int \frac{cos ( 8 x )}{1 + sin ^{2} ( 8 x )} d x

Apply u-substitution

: \int \frac{cos ( u )}{8 ( 1 + sin ^{2} ( u ) )} d u

= \int \frac{cos ( u )}{8 ( 1 + sin ^{2} ( u ) )} d u

Take the constant out:

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

= \frac{1}{8} \cdot \int \frac{cos ( u )}{1 + sin ^{2} ( u )} d u

Apply u-substitution

: \int \frac{1}{1 + v ^{2}} d v

= \frac{1}{8} \cdot \int \frac{1}{1 + v ^{2}} d v

Use the common integral:

\int \frac{1}{1 + v ^{2}} d v = arctan (v)

= \frac{1}{8} arctan (v)

Substitute back

= \frac{1}{8} arctan (sin (8 x))

Add a constant to the solution

= \frac{1}{8} arctan (sin (8 x)) + C

\int \frac{x}{1 + 3 x ^{2}} d x

\frac{d}{d y} (x^{2} + 4 x y + y^{6} - 8)

\frac{d}{d x} (x^{3} + 6 x^{2} + 6 x)

f (x) = 2 - 5 x^{2}

\frac{3 s}{( s + 2 ) ( s - 4 )}

What is the integral of (cos(8x))/(1+sin^2(8x)) ?
The integral of (cos(8x))/(1+sin^2(8x)) is 1/8 arctan(sin(8x))+C