MHT CET · Maths · Indefinite Integration
If \(\int \frac{\sqrt{x}}{x(x+1)} d x=k \tan ^{-1} m+c\), (where \(c\) is constant of integration), then
- A \(\mathrm{k}=1, \mathrm{~m}=\sqrt{\mathrm{x}}\)
- B \(\mathrm{k}=2, \mathrm{~m}=\sqrt{\mathrm{x}}\)
- C \(\mathrm{k}=1, \mathrm{~m}=\mathrm{x}\)
- D \(\mathrm{k}=2, \mathrm{~m}=\mathrm{x}\)
Answer & Solution
Correct Answer
(B) \(\mathrm{k}=2, \mathrm{~m}=\sqrt{\mathrm{x}}\)
Step-by-step Solution
Detailed explanation
\(
I=\int \frac{\sqrt{x}}{x(x+1)} d x
\)
Put \(x \tan ^2 \theta \Rightarrow d x=2 \tan \theta \sec ^2 \theta d \theta\)
\(\therefore I=\int \frac{\tan \theta\left(2 \tan \theta \sec ^2 \theta\right)}{\tan ^2 \theta(1+\tan \theta)} d \theta \)
\( =2 \int \frac{\sec ^2 \theta}{\sec ^2 \theta} d \theta=2 \int d \theta=2 \theta \)
\( =2 \tan ^{-1} \sqrt{x}+c\)
Comparing with given data, \(\mathrm{k}=2, \mathrm{~m}=\sqrt{\mathrm{x}}\)
I=\int \frac{\sqrt{x}}{x(x+1)} d x
\)
Put \(x \tan ^2 \theta \Rightarrow d x=2 \tan \theta \sec ^2 \theta d \theta\)
\(\therefore I=\int \frac{\tan \theta\left(2 \tan \theta \sec ^2 \theta\right)}{\tan ^2 \theta(1+\tan \theta)} d \theta \)
\( =2 \int \frac{\sec ^2 \theta}{\sec ^2 \theta} d \theta=2 \int d \theta=2 \theta \)
\( =2 \tan ^{-1} \sqrt{x}+c\)
Comparing with given data, \(\mathrm{k}=2, \mathrm{~m}=\sqrt{\mathrm{x}}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- The region represented by the inequation system \(x, y \geq 0, y \leq 6, x+y \leq 3\), isMHT CET 2008 Easy
- The value of \(\sec ^2\left(\tan ^{-1} 2\right)+\operatorname{cosec}^2\left(\cot ^{-1} 3\right)\) isMHT CET 2023 Easy
- If \(\mathrm{p} \rightarrow(\sim \mathrm{p} \vee \sim \mathrm{q})\) is false, then the truth values of \(p\) and \(q\) are respectivelyMHT CET 2024 Easy
- If \(\mathrm{f}(x)= \begin{cases}-2 \sin x & , \quad x \leqslant-\frac{\pi}{2} \\ a \sin x+\mathrm{b} & , \quad \frac{-\pi}{2} < x < \frac{\pi}{2} \\ \cos x & , \quad x \geqslant \frac{\pi}{2}\end{cases}\) is continuous at \(\mathrm{x}=\frac{-\pi}{2}\) and \(x=\frac{\pi}{2}\), then the value of \(2 a+\mathrm{b}\) isMHT CET 2025 Medium
- Given \(\mathrm{A}=\left[\begin{array}{lll}x & 3 & 2 \\ 1 & y & 4 \\ 2 & 2 & z\end{array}\right], x y z=60\) and \(8 x+4 y+3 z=20\), then \(\mathrm{A} \cdot(\operatorname{adjA})\) is equal toMHT CET 2022 Medium
- Which of the following is NOT equivalent toMHT CET 2019 Easy
More PYQs from MHT CET
- If the vector \(\bar{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}, \bar{b}=-\hat{i}+2 \hat{j}+\hat{k}\) and \(\bar{c}=3 \hat{i}+\hat{j}\) are such that \((\bar{a}+\lambda \bar{b})\) is perpendicular to \(\bar{c}\), then the value of \(\lambda\) isMHT CET 2022 Easy
- Which among the following carbohydrates is a trisaccharide?MHT CET 2021 Medium
- The frequency of vibrating air column in a pipe, open at both ends is \(f_1\). When \(\left(\frac{3}{4}\right)^{\text {th }}\) of its length is immersed in water, the frequency of vibrating air column is \(f_2\). The value of \(\frac{f_1}{f_2}\) isMHT CET 2022 Medium
- In the process of space communication, use of modem is necessary.
In which one of the following modes modem acts as a modulator and a demodulator respectively?MHT CET 2020 Easy - The existence of electromagnetic waves were experimentally confirmed byMHT CET 2012 Easy
- Formation of is explained on the basis of what hybridisation of phosphorus atom?MHT CET 2018 Easy