What is the integral of e^{3x}x+e^x ?

The integral of e^{3x}x+e^x is 1/9 (e^{3x}*3x-e^{3x})+e^x+C

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integral of e^{3x}x+e^x

\int e^{3 x} x + e^{x} d x

\frac{1}{9} (e^{3 x} \cdot 3 x - e^{3 x}) + e^{x} + C

Solution steps

\int e^{3 x} x + e^{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 e^{3 x} x d x + \int e^{x} d x

\int e^{3 x} x d x = \frac{1}{9} (e^{3 x} \cdot 3 x - e^{3 x})

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

= \frac{1}{9} (e^{3 x} \cdot 3 x - e^{3 x}) + e^{x}

Add a constant to the solution

= \frac{1}{9} (e^{3 x} \cdot 3 x - e^{3 x}) + e^{x} + C

\int \frac{x}{27 x ^{2} - 1} d x

x \to 0 + lim (- x + 3)

\frac{d ^{2} y}{d x ^{2}} + 25 y = 0

x \to 1 - lim (\frac{1 - x}{1 + x})

\frac{8 x ^{2} + 8 x + 6}{x}

What is the integral of e^{3x}x+e^x ?
The integral of e^{3x}x+e^x is 1/9 (e^{3x}*3x-e^{3x})+e^x+C