What is the integral of (cos(ax))/(sqrt(9+sin(ax))) ?

The integral of (cos(ax))/(sqrt(9+sin(ax))) is 2/a sqrt(9+sin(ax))+C

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integral of (cos(ax))/(sqrt(9+sin(ax)))

\int \frac{cos ( a x )}{9 + sin ( a x )} d x

\frac{2}{a} 9 + sin (a x) + C

Solution steps

\int \frac{cos ( a x )}{9 + sin ( a x )} d x

Apply u-substitution

: \int \frac{cos ( u )}{a 9 + sin ( u )} d u

= \int \frac{cos ( u )}{a 9 + sin ( u )} d u

Take the constant out:

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

= \frac{1}{a} \cdot \int \frac{cos ( u )}{9 + sin ( u )} d u

Apply u-substitution

: \int \frac{1}{v} d v

= \frac{1}{a} \cdot \int \frac{1}{v} d v

Apply the Power Rule

: 2 v^{\frac{1}{2}}

= \frac{1}{a} \cdot 2 v^{\frac{1}{2}}

Substitute back

= \frac{1}{a} \cdot 2 (9 + sin (a x))^{\frac{1}{2}}

Simplify

\frac{1}{a} \cdot 2 (9 + sin (a x))^{\frac{1}{2}} : \frac{2}{a} 9 + sin (a x)

= \frac{2}{a} 9 + sin (a x)

Add a constant to the solution

= \frac{2}{a} 9 + sin (a x) + C

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

f (x) = x + ln (x)

\int_{1}^{\infty} x d x

y = x^{2} + 3 x - 5, at x = 2

(e^{x} + y) d x + (e^{v} + x) d y = 0

What is the integral of (cos(ax))/(sqrt(9+sin(ax))) ?
The integral of (cos(ax))/(sqrt(9+sin(ax))) is 2/a sqrt(9+sin(ax))+C