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

The integral of e^{2x}*x is 1/4 (e^{2x}*2x-e^{2x})+C

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integral of e^{2x}*x

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

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

Solution steps

\int e^{2 x} x d x

Apply u-substitution

: \int \frac{e ^{u} u}{4} d u

= \int \frac{e ^{u} u}{4} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{4} \cdot \int e^{u} u d u

Apply Integration By Parts

: e^{u} u - \int e^{u} d u

= \frac{1}{4} (e^{u} u - \int e^{u} d u)

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

= \frac{1}{4} (e^{u} u - e^{u})

Substitute back

u = 2 x

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

Add a constant to the solution

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

y^{^{'}} = 0.05 ln (\frac{4000}{y}) y

ln (1 + 2 x)

\frac{d}{d x} (6 x^{2} - 6 x - 12)

(ln (x))^{3}

\int \frac{2 t}{1 - 4 t ^{4}} d t

What is the integral of e^{2x}*x ?
The integral of e^{2x}*x is 1/4 (e^{2x}*2x-e^{2x})+C