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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2018 · Mathematics · Indefinite Integration

What is ∫ ^3 x x d x equal to? Where c is the constant o...

What is ∫sin⁡3xcos⁡xdx\int \sin ^3 x \cos x d x equal to? Where c is the constant o integration.

Options

  1. Acos⁡4x+c\cos ^4 \mathrm{x}+\mathrm{c}
  2. Bsin⁡4x+c\sin ^4 x+c
  3. C(1−sin⁡2x)24+c\frac{\left(1-\sin ^2 x\right)^2}{4}+c
  4. D(1−cos⁡2x)24+c\frac{\left(1-\cos ^2 x\right)^2}{4}+c

Correct answer

D. (1−cos⁡2x)24+c\frac{\left(1-\cos ^2 x\right)^2}{4}+c

Step-by-step solution

Let t=sin⁡x⇒dt=cos⁡xdxt=\sin x \Rightarrow d t=\cos x d x∫sin⁡3xcos⁡xdx=∫t3dt=t44+c=sin⁡4x4+c=(1−cos⁡2x)24+c\begin{aligned} & \int \sin ^3 x \cos x d x=\int t^3 d t=\frac{t^4}{4}+c=\frac{\sin ^4 x}{4}+c \\ & =\frac{\left(1-\cos ^2 x\right)^2}{4}+c \end{aligned}$$
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