MHT CET · Maths · Definite Integration
The value of \(\int_{0}^{\pi / 2} \log (\operatorname{cosec} x) d x\) is
- A \(\frac{\pi}{2} \log 2\)
- B \(\pi \log 2\)
- C \(-\frac{\pi}{2} \log 2\)
- D \(2 \pi \log 2\)
Answer & Solution
Correct Answer
(A) \(\frac{\pi}{2} \log 2\)
Step-by-step Solution
Detailed explanation
Let \(\quad I=\int_{0}^{\pi / 2} \log (\operatorname{cosec} x) d x\)
\(=\int_{0}^{\pi / 2} \log \left(\frac{1}{\sin x}\right) d x\)
\(=-\int_{0}^{\pi / 2} \log \sin x d x\)
\(=\frac{\pi}{2} \log 2\)
\(\left[\because \int_{0}^{\pi / 2} \log \sin x d x=-\frac{\pi}{2} \log 2\right]\)
\(=\int_{0}^{\pi / 2} \log \left(\frac{1}{\sin x}\right) d x\)
\(=-\int_{0}^{\pi / 2} \log \sin x d x\)
\(=\frac{\pi}{2} \log 2\)
\(\left[\because \int_{0}^{\pi / 2} \log \sin x d x=-\frac{\pi}{2} \log 2\right]\)
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
- Let \(\mathrm{A}=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right], x \in \mathbb{R}^{+}\)and \(\mathrm{A}^4=\left[\mathrm{a}_{\mathrm{ij}}\right]_2\).
If \(\mathrm{a}_{11}=109\), then \(\left(\mathrm{A}^4\right)^{-1}=\)MHT CET 2024 Medium - \(\int_0^{\frac{\pi}{2}} \frac{300 \sin x+100 \cos x}{\sin x+\cos x} \mathrm{~d} x=\ldots\)MHT CET 2025 Medium
- If \(\lim _{x \rightarrow 1} \frac{x^2-\mathrm{a} x+\mathrm{b}}{x-1}=7\), then \(\mathrm{a}+\mathrm{b}\) is equal toMHT CET 2024 Medium
- If \(\mathrm{f}(x)=\cot ^{-1}\left(\frac{x^x-x^{-x}}{2}\right)\), then the value of \(\mathrm{f}^{\prime}(1)\) is equal toMHT CET 2025 Medium
- \(\int \frac{x}{\sqrt{1-2 x^4}} \mathrm{~d} x=\) (Where \(C\) is a constant of integration)MHT CET 2022 Medium
- If \(A=\left[\begin{array}{ll}3 & 2 \\ 0 & 1\end{array}\right]\), then \(\left(A^{-1}\right)^3=\)MHT CET 2022 Easy
More PYQs from MHT CET
- A condenser of capacity ' \(\mathrm{C}\) ' is charged to a potential difference of ' \(\mathrm{V}_1\) '. The plates of the condenser are then connected to an ideal inductor of inductance ' \(L\) '. The current through an inductor |when the potential difference across the condenser reduces to ' \(\mathrm{V}\) ' isMHT CET 2022 Hard
- In a triangle \(A B C\), with usual notations.
\(\frac{2 \cos A}{a}+\frac{\cos B}{b}+\frac{2 \cos C}{c}=\frac{a}{b c}+\frac{b}{c a}\)
Then \(\angle \mathrm{A}=\)MHT CET 2025 Medium - According to kinetic theory of gases, when two molecules of a gas collide with each other thenMHT CET 2022 Easy
- The value of \(\int_0^1 \tan ^{-1}\left(1-x+x^2\right) \mathrm{d} x\) isMHT CET 2025 Medium
- A pair of tangents are drawn to the circle \(x^2+y^2+6 x-4 y-12=0\) from a point \(\mathrm{P}(-4,-5)\), then the area enclosed between these tangents and the area of the circle isMHT CET 2025 Medium
- Identify the substrate ' \(A\) ' in the following conversion.
\(\mathrm{A} \xrightarrow[\mathrm{H}_3 \mathrm{O}^{+}]{\mathrm{AlH}(\mathrm{i}-\mathrm{Bu})_2} \text { Pent-3-enal }\)MHT CET 2025 Medium