What is the integral of (2e^x+3sin(x))^2 ?

The integral of (2e^x+3sin(x))^2 is 2e^{2x}+12(-(e^xcos(x))/2+(e^xsin(x))/2)+9/2 (x-1/2 sin(2x))+C

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integral of (2e^x+3sin(x))^2

\int (2 e^{x} + 3 sin (x))^{2} d x

2 e^{2 x} + 12 (- \frac{e ^{x} cos ( x )}{2} + \frac{e ^{x} sin ( x )}{2}) + \frac{9}{2} (x - \frac{1}{2} sin (2 x)) + C

Solution steps

\int (2 e^{x} + 3 sin (x))^{2} d x

Expand

(2 e^{x} + 3 sin (x))^{2} : 4 e^{2 x} + 12 e^{x} sin (x) + 9 sin^{2} (x)

= \int 4 e^{2 x} + 12 e^{x} sin (x) + 9 sin^{2} (x) 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 4 e^{2 x} d x + \int 12 e^{x} sin (x) d x + \int 9 sin^{2} (x) d x

\int 4 e^{2 x} d x = 2 e^{2 x}

\int 12 e^{x} sin (x) d x = 12 (- \frac{e ^{x} cos ( x )}{2} + \frac{e ^{x} sin ( x )}{2})

\int 9 sin^{2} (x) d x = \frac{9}{2} (x - \frac{1}{2} sin (2 x))

= 2 e^{2 x} + 12 (- \frac{e ^{x} cos ( x )}{2} + \frac{e ^{x} sin ( x )}{2}) + \frac{9}{2} (x - \frac{1}{2} sin (2 x))

Add a constant to the solution

= 2 e^{2 x} + 12 (- \frac{e ^{x} cos ( x )}{2} + \frac{e ^{x} sin ( x )}{2}) + \frac{9}{2} (x - \frac{1}{2} sin (2 x)) + C

\int \frac{1000}{1 + 3 x} d x

\int 3 - 5 x d x

\int 3 sin (2 x) sin (x) d x

\int \frac{x ^{2} - 2 x + 1}{x ^{2} - 2 x + 50} d x

What is the integral of (2e^x+3sin(x))^2 ?
The integral of (2e^x+3sin(x))^2 is 2e^{2x}+12(-(e^xcos(x))/2+(e^xsin(x))/2)+9/2 (x-1/2 sin(2x))+C