What is the integral of (2x+15)e^{0.25x} ?

The integral of (2x+15)e^{0.25x} is 8e^{0.25x}x+28e^{0.25x}+C

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integral of (2x+15)e^{0.25x}

\int (2 x + 15) e^{0.25 x} d x

8 e^{0.25 x} x + 28 e^{0.25 x} + C

Solution steps

\int (2 x + 15) e^{0.25 x} d x

Expand

(2 x + 15) e^{0.25 x} : 2 e^{0.25 x} x + 15 e^{0.25 x}

= \int 2 e^{0.25 x} x + 15 e^{0.25 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 2 e^{0.25 x} x d x + \int 15 e^{0.25 x} d x

\int 2 e^{0.25 x} x d x = 2 (4 e^{0.25 x} x - 16 e^{0.25 x})

\int 15 e^{0.25 x} d x = 60 e^{0.25 x}

= 2 (4 e^{0.25 x} x - 16 e^{0.25 x}) + 60 e^{0.25 x}

Simplify

2 (4 e^{0.25 x} x - 16 e^{0.25 x}) + 60 e^{0.25 x} : 8 e^{0.25 x} x + 28 e^{0.25 x}

= 8 e^{0.25 x} x + 28 e^{0.25 x}

Add a constant to the solution

= 8 e^{0.25 x} x + 28 e^{0.25 x} + C

x \to \infty lim (1^{0})

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What is the integral of (2x+15)e^{0.25x} ?
The integral of (2x+15)e^{0.25x} is 8e^{0.25x}x+28e^{0.25x}+C