What is the integral of xe^{5x} ?

The integral of xe^{5x} is 1/25 (e^{5x}*5x-e^{5x})+C

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integral of xe^{5x}

\int x e^{5 x} d x

\frac{1}{25} (e^{5 x} \cdot 5 x - e^{5 x}) + C

Solution steps

\int x e^{5 x} d x

Apply u-substitution

: \int \frac{e ^{u} u}{25} d u

= \int \frac{e ^{u} u}{25} d u

Take the constant out:

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

= \frac{1}{25} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= \frac{1}{25} (e^{u} u - \int e^{u} d u)

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

= \frac{1}{25} (e^{u} u - e^{u})

Substitute back

u = 5 x

= \frac{1}{25} (e^{5 x} \cdot 5 x - e^{5 x})

Add a constant to the solution

= \frac{1}{25} (e^{5 x} \cdot 5 x - e^{5 x}) + C

h \to 0 lim (\frac{1}{h ^{2}})

\int x n (x) d x

\int sec^{13} (x) d x

f (x) = sinh (2 x)

\int_{0}^{l n (2)} e^{x + 2 y} d x

What is the integral of xe^{5x} ?
The integral of xe^{5x} is 1/25 (e^{5x}*5x-e^{5x})+C