What is the integral of (9^x-12e^x-5sin(x)+8) ?

The integral of (9^x-12e^x-5sin(x)+8) is (9^x)/(2ln(3))-12e^x+5cos(x)+8x+C

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integral of (9^x-12e^x-5sin(x)+8)

\int (9^{x} - 12 e^{x} - 5 sin (x) + 8) d x

\frac{9 ^{x}}{2 ln ( 3 )} - 12 e^{x} + 5 cos (x) + 8 x + C

Solution steps

\int 9^{x} - 12 e^{x} - 5 sin (x) + 8 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 9^{x} d x - \int 12 e^{x} d x - \int 5 sin (x) d x + \int 8 d x

\int 9^{x} d x = \frac{9 ^{x}}{2 ln ( 3 )}

\int 12 e^{x} d x = 12 e^{x}

\int 5 sin (x) d x = - 5 cos (x)

\int 8 d x = 8 x

= \frac{9 ^{x}}{2 ln ( 3 )} - 12 e^{x} - (- 5 cos (x)) + 8 x

Simplify

= \frac{9 ^{x}}{2 ln ( 3 )} - 12 e^{x} + 5 cos (x) + 8 x

Add a constant to the solution

= \frac{9 ^{x}}{2 ln ( 3 )} - 12 e^{x} + 5 cos (x) + 8 x + C

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What is the integral of (9^x-12e^x-5sin(x)+8) ?
The integral of (9^x-12e^x-5sin(x)+8) is (9^x)/(2ln(3))-12e^x+5cos(x)+8x+C