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BITSAT 2023Paper 2
Exam
BITSAT
Year
2023
Paper
Paper 2
Subject
Mathematics
Chapter
Indefinite Integration

BITSAT 2023 · Paper 2 · Mathematics · Indefinite Integration

Value of ∫ d x x(a-x) is

Value of ∫dxx(a−x)\int \frac{d x}{\sqrt{x(a-x)}} is

Options

  1. A2sin⁡−1xa+c2 \sin ^{-1} \sqrt{\frac{x}{a}}+c
  2. B−2sin⁡−1xa+c-2 \sin ^{-1} \sqrt{\frac{x}{a}}+c
  3. C2sin⁡−1xa+c2 \sin ^{-1} \frac{\sqrt{x}}{a}+c
  4. D
    None of these

Correct answer

A. 2sin⁡−1xa+c2 \sin ^{-1} \sqrt{\frac{x}{a}}+c

Step-by-step solution

Let x=asin⁡2θ\mathrm{x}=\mathrm{a} \sin ^2 \theta
then dx=2asin⁡θcos⁡θdθd x=2 a \sin \theta \cos \theta d \theta
∴I=∫2asin⁡θ⋅cos⁡θasin⁡2θ⋅acos⁡2θdθ=2∫dθ=2θ+c=2sin⁡−1(x/a)+c\begin{aligned} \\ \therefore \quad I & =\int \frac{2 a \sin \theta \cdot \cos \theta}{\sqrt{a \sin ^2 \theta \cdot a \cos ^2 \theta}} d \theta \\ \\ & =2 \int d \theta=2 \theta+c \\ \\ & =2 \sin ^{-1}(\sqrt{x / a})+c \\ \end{aligned}
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