What is the integral of 9^{1/2 x+sin(x)}(1+2cos(x)) ?

The integral of 9^{1/2 x+sin(x)}(1+2cos(x)) is (3^{x+2sin(x)})/(ln(3))+C

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integral of 9^{1/2 x+sin(x)}(1+2cos(x))

\int 9^{\frac{1}{2} x + s i n (x)} (1 + 2 cos (x)) d x

\frac{3 ^{x + 2 s i n (x)}}{ln ( 3 )} + C

Solution steps

\int 9^{\frac{1}{2} x + s i n (x)} (1 + 2 cos (x)) d x

Apply u-substitution

: \int 2 \cdot 9^{u} d u

= \int 2 \cdot 9^{u} d u

Take the constant out:

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

= 2 \cdot \int 9^{u} d u

\int a^{x} d x = \frac{a ^{x}}{ln a}

= 2 \cdot \frac{9 ^{u}}{ln ( 9 )}

Substitute back

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

= 2 \cdot \frac{9 ^{\frac{1}{2} x + s i n (x)}}{ln ( 9 )}

Simplify

2 \cdot \frac{9 ^{\frac{1}{2} x + s i n (x)}}{ln ( 9 )} : \frac{3 ^{x + 2 s i n (x)}}{ln ( 3 )}

= \frac{3 ^{x + 2 s i n (x)}}{ln ( 3 )}

Add a constant to the solution

= \frac{3 ^{x + 2 s i n (x)}}{ln ( 3 )} + C

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What is the integral of 9^{1/2 x+sin(x)}(1+2cos(x)) ?
The integral of 9^{1/2 x+sin(x)}(1+2cos(x)) is (3^{x+2sin(x)})/(ln(3))+C