What is the integral of (2+5x)e^{1/3 x} ?

The integral of (2+5x)e^{1/3 x} is 15e^{x/3}x-39e^{x/3}+C

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integral of (2+5x)e^{1/3 x}

\int (2 + 5 x) e^{\frac{1}{3} x} d x

15 e^{\frac{x}{3}} x - 39 e^{\frac{x}{3}} + C

Solution steps

\int (2 + 5 x) e^{\frac{1}{3} x} d x

Expand

(2 + 5 x) e^{\frac{1}{3} x} : 2 e^{\frac{x}{3}} + 5 e^{\frac{x}{3}} x

= \int 2 e^{\frac{x}{3}} + 5 e^{\frac{x}{3}} 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^{\frac{x}{3}} d x + \int 5 e^{\frac{x}{3}} x d x

\int 2 e^{\frac{x}{3}} d x = 6 e^{\frac{x}{3}}

\int 5 e^{\frac{x}{3}} x d x = 5 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}})

= 6 e^{\frac{x}{3}} + 5 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}})

Simplify

6 e^{\frac{x}{3}} + 5 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}}) : 15 e^{\frac{x}{3}} x - 39 e^{\frac{x}{3}}

= 15 e^{\frac{x}{3}} x - 39 e^{\frac{x}{3}}

Add a constant to the solution

= 15 e^{\frac{x}{3}} x - 39 e^{\frac{x}{3}} + C

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What is the integral of (2+5x)e^{1/3 x} ?
The integral of (2+5x)e^{1/3 x} is 15e^{x/3}x-39e^{x/3}+C