What is the integral of e^{2x}-xe^{2x} ?

The integral of e^{2x}-xe^{2x} is -1/2 e^{2x}x+3/4 e^{2x}+C

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integral of e^{2x}-xe^{2x}

\int e^{2 x} - x e^{2 x} d x

- \frac{1}{2} e^{2 x} x + \frac{3}{4} e^{2 x} + C

Solution steps

\int e^{2 x} - x 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} d x - \int x e^{2 x} d x

\int e^{2 x} d x = \frac{1}{2} e^{2 x}

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

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

Simplify

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

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

Add a constant to the solution

= - \frac{1}{2} e^{2 x} x + \frac{3}{4} e^{2 x} + C

\frac{d}{d x} (- 7 csc (x) + \frac{1}{x ^{5}})

arcsin (\frac{x}{a})

(\frac{4}{3}) π x^{3}

\frac{d}{d x} (4 x y^{2} - x^{2} y^{2} - x y^{3})

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

What is the integral of e^{2x}-xe^{2x} ?
The integral of e^{2x}-xe^{2x} is -1/2 e^{2x}x+3/4 e^{2x}+C