What is the integral of e^{2x}x+3/2 e^{2x} ?

The integral of e^{2x}x+3/2 e^{2x} is 1/2 e^{2x}x+1/2 e^{2x}+C

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integral of e^{2x}x+3/2 e^{2x}

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

\frac{1}{2} e^{2 x} x + \frac{1}{2} e^{2 x} + C

Solution steps

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

\int e^{2 x} x d x = \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x})

\int \frac{3}{2} e^{2 x} d x = \frac{3}{4} e^{2 x}

= \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}) + \frac{3}{4} e^{2 x}

Simplify

\frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}) + \frac{3}{4} e^{2 x} : \frac{1}{2} e^{2 x} x + \frac{1}{2} e^{2 x}

= \frac{1}{2} e^{2 x} x + \frac{1}{2} e^{2 x}

Add a constant to the solution

= \frac{1}{2} e^{2 x} x + \frac{1}{2} e^{2 x} + C

n = 1 \sum \infty \frac{3 n}{5 n + 1}

2 x, 2 x - 4, 0, 4

\int - 2 t^{2} d t

\frac{\partial}{\partial x} (cos (4 x + 5 y) \cdot 5)

\frac{5}{2 s ^{2} + 3 s + 1}

What is the integral of e^{2x}x+3/2 e^{2x} ?
The integral of e^{2x}x+3/2 e^{2x} is 1/2 e^{2x}x+1/2 e^{2x}+C