MHT CET · Maths · Indefinite Integration
\(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \frac{\operatorname{cosec} x \cdot \cot x}{1+\operatorname{cosec}^2 x} d x=\)
- A \(\frac{\pi}{4}-\tan ^{-1} 2\)
- B \(\tan ^{-1} 1\)
- C \(\tan ^{-1} 2\)
- D \(\tan ^{-1}\left(\frac{1}{3}\right)\)
Answer & Solution
Correct Answer
(D) \(\tan ^{-1}\left(\frac{1}{3}\right)\)
Step-by-step Solution
Detailed explanation
Let \(I=\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \frac{\operatorname{cosec} x \cdot \cot x}{1+\operatorname{cosec}^2 x} d x\)
Put \(\operatorname{cosec} x=t \Rightarrow \operatorname{cosec} x \cot x=-d t\). When \(x=\frac{\pi}{6}, t=2\) and when \(x=\frac{\pi}{2}, t=1\)
\(\mathrm{I}=\int_2^1(-1) \frac{\mathrm{dt}}{1+\mathrm{t}^2} \)
\( =\int_1^2 \frac{\mathrm{dt}}{1+\mathrm{t}^2}=[\tan ^{-1} \mathrm{t}_1^2=\tan ^{-1}(2)-\tan ^{-1}\) \((1)=\tan ^{-1}\left[\frac{2-1}{1+(2)(1)}\right]=\tan ^{-1}\left(\frac{1}{3}\right)\)
Put \(\operatorname{cosec} x=t \Rightarrow \operatorname{cosec} x \cot x=-d t\). When \(x=\frac{\pi}{6}, t=2\) and when \(x=\frac{\pi}{2}, t=1\)
\(\mathrm{I}=\int_2^1(-1) \frac{\mathrm{dt}}{1+\mathrm{t}^2} \)
\( =\int_1^2 \frac{\mathrm{dt}}{1+\mathrm{t}^2}=[\tan ^{-1} \mathrm{t}_1^2=\tan ^{-1}(2)-\tan ^{-1}\) \((1)=\tan ^{-1}\left[\frac{2-1}{1+(2)(1)}\right]=\tan ^{-1}\left(\frac{1}{3}\right)\)
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 logical expression
\([\mathrm{p} \wedge(\mathrm{q} \vee \mathrm{r})] \vee[(\sim \mathrm{p} \wedge \mathrm{q}) \vee(\sim \mathrm{p} \wedge \mathrm{r})]\) is equivalent toMHT CET 2020 Medium - The general solution of \(\tan 3 x=1\) isMHT CET 2020 Easy
- \(\int_{0}^{\pi} \frac{x d x}{1+\cos \alpha \sin x},(0 < \alpha < \pi)\) is equal toMHT CET 2011 Hard
- The p.m.f. of a random variable \(\mathrm{X}\) is \(\mathrm{P}(x)=\left\{\begin{array}{cl}\frac{2 x}{\mathrm{n}(\mathrm{n}+1)} & , \quad x=1,2,3, \ldots \mathrm{n} \ 0 & , \text { otherwise }\end{array}\right.\), then \(\mathrm{E}(\mathrm{X})\) isMHT CET 2023 Medium
- If \(u=\cos ^3 x, v=\sin ^3 x\), then \(\left(\frac{d v}{d u}\right)_{x=\frac{\pi}{4}}\) is equal toMHT CET 2021 Easy
- The tangent to the curve \(y=x^3+a x-b\) at the point \((1,-5)\) is perpendicular to the line \(y-x+4=0\), then which one of the following points lies on the curve?MHT CET 2022 Medium
More PYQs from MHT CET
- The co-ordinates of the foot of the perpendicular from the point \((0,2,3)\) on the line
\(\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}\) areMHT CET 2020 Easy - Light of wavelength ' \(\lambda\) ' falls on a metal having work function \(\frac{\mathrm{hc}}{\lambda_0}\). Photoelectric effect will take place only if ( \(\lambda_0\) is the threshold wavelength)MHT CET 2025 Easy
- If \(A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]\) and \(X\) is a \(2 \times 2\) matrix such that \(A X=1\), then \(X=\)MHT CET 2020 Easy
- In a class of 300 students, every student reads 5 news papers and every news paper is read by 60 students. Then the number of newspapers isMHT CET 2024 Easy
- Pure Silicon crystal at \(300 \mathrm{~K}\) has equal electron and hole concentration of \(1.5 \times 10^{16} \mathrm{~m}^{-3}\). Doping by indium increases \(n_h=4.5 \times 10^{22} \mathrm{~m}^{-3}\). The \(n_e\) in the doped silicon is:MHT CET 2022 Easy
- Which from following reagents is used in the conversion of phenol to picric acid?MHT CET 2024 Medium