First slide

Methods of integration

Question

$\int x^{x} \ln (e x) d x is equal to$

Moderate

A

$x^{x} + c$
B

$x . \ln x + c$
C

${(\ln x)}^{x} + c$
D

$x^{\ln x} + c$

Solution

$\begin{array}{l} I = \int x^{x} \ln (e x) d x = \int x^{x} (1 + \ln x) d x \\ let t = x^{x} = e^{x \ln x} \Rightarrow \frac{d t}{d x} = x^{x} (x \cdot \frac{1}{x} + \ln x) = x^{x} (1 + \ln x) \\ \Rightarrow d t = x^{x} (1 + \ln x) d x \\ ∴ I = \int d t = t + C = x^{x} + C \end{array}$

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