Question

Find the integral with respect to
x:∫(exx)(ex1)dx

Answer 1

Answer on Question #45967 – Math - Integral Calculus

Find the integral with respect to

x:J(exx)(ex1)dx

Solution

$$\begin{array}{l} \int x \cdot e ^ {x} \cdot e ^ {x + 1} d x = e \int x \cdot e ^ {2 x} d x = e \int x \frac {d (e ^ {2 x})}{2} = e \left(\frac {x e ^ {2 x}}{2} - \frac {1}{2} \int e ^ {2 x} d x\right) \\ = \frac {e}{2} \left(x e ^ {2 x} - \frac {e ^ {2 x}}{2} + C\right) = \frac {e}{2} \left(x e ^ {2 x} - \frac {e ^ {2 x}}{2}\right) + C _ {0} \end{array}$$

In this task the next rule (integration by parts) were used :

$$\int u \cdot d v = u \cdot v - \int v \cdot d u$$

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