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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2022 · Mathematics · Indefinite Integration

What is ∫(x^x)^2(1+ x) d x equal to?

What is ∫(xx)2(1+ln⁡x)dx\int\left(x^x\right)^2(1+\ln x) d x equal to?

Options

  1. Ax2x+cx^{2 \mathrm{x}}+c
  2. B12x2x+c\frac{1}{2} x^{2 \mathrm{x}}+c
  3. C2x2x+c2 x^{2 \mathrm{x}}+c
  4. D12xx+c\frac{1}{2} x^x+c

Correct answer

B. 12x2x+c\frac{1}{2} x^{2 \mathrm{x}}+c

Step-by-step solution

I=∫(xx)2(1+∫ln⁡x)dxI=\int\left(x^x\right)^2\left(1+\int \ln x\right) d x Let xx=tx^{\mathrm{x}}=t⇒ln⁡t=xln⁡x⇒dt=xx(1+ln⁡x)dxI=∫tdt=−2+C=−(x)+C12x2x+C\begin{aligned} & \Rightarrow \ln t=x \ln x \Rightarrow d t=x^{\mathrm{x}}(1+\ln x) d x \\ & I=\int t d t=\frac{-}{2}+C=-(x)+C \\ & \frac{1}{2} x^{2 \mathrm{x}}+C \end{aligned}$$
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