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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Definite Integration

NDA 2024 · Mathematics · Definite Integration

What is ∫_ -1 ^1(3 x- 3 x) ^2 x d x equal to?

What is ∫−11(3sin⁡x−sin⁡3x)cos⁡2xdx\int_{-1}^1(3 \sin x-\sin 3 x) \cos ^2 x d x equal to?

Options

  1. A−14-\frac{1}{4}
  2. B0
  3. C12\frac{1}{2}
  4. D14\frac{1}{4}

Correct answer

B. 0

Step-by-step solution

Let, I=∫−11(3sin⁡x−sin⁡3x)cos⁡2xdx\mathrm{I}=\int_{-1}^1(3 \sin x-\sin 3 x) \cos ^2 x d xI=∫−11(3sin⁡x−3sin⁡x+4sin⁡3x)cos⁡2xdxI=∫−114sin⁡3xcos⁡2xdx=∫−11f(x)dx\begin{aligned} \mathrm{I} & =\int_{-1}^1\left(3 \sin x-3 \sin x+4 \sin ^3 x\right) \cos ^2 x d x \\ I & =\int_{-1}^1 4 \sin ^3 x \cos ^2 x d x \\ & =\int_{-1}^1 f(x) d x \end{aligned}$Here,Here,f(x)=4 \sin ^3 x \cos ^2 xisanoddfunctionasis an odd function as\quad f(-x)=-f(x)\Rightarrow \quad I\mathrm{I}=\int_{-1}^1 4 \sin ^2 x \cos ^2 x d x=0$
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