MHT CET · Maths · Definite Integration
\(\int_0^{\pi / 4} \log (1+\tan x) d x=\)
- A \(\frac{\pi}{16} \log 2\)
- B \(\frac{\pi}{4} \log 2\)
- C \(\frac{\pi}{8} \log 2\)
- D \(\pi \log 2\)
Answer & Solution
Correct Answer
(C) \(\frac{\pi}{8} \log 2\)
Step-by-step Solution
Detailed explanation
\( \text { Let } I=\int_0^{\pi / 4} \log (1+\tan x) d x \)
\( =\int_0^{\pi / 4} \log \left[1+\tan \left(\frac{\pi}{4}-x\right)\right] d x \)
\( =\int_0^{\pi / 4} \log \left[1+\left(\frac{1-\tan x}{1+\tan x}\right)\right] d x=\int_0^{\pi / 4}\) \(\log \left(\frac{2}{1+\tan x}\right) d x \)
\( =\int_0^{\pi / 4}(\log 2) d x-\int_0^{\pi / 4} \log (1+\tan x) d x=\) \(\int_0^{\pi / 4}(\log 2) d x-I \)
\( \therefore 2 I=(\log 2)[x]_0^{\pi / 4}=(\log 2)\left(\frac{\pi}{4}\right) \Rightarrow I=\left(\frac{\pi}{8}\right) \log 2\)
\( =\int_0^{\pi / 4} \log \left[1+\tan \left(\frac{\pi}{4}-x\right)\right] d x \)
\( =\int_0^{\pi / 4} \log \left[1+\left(\frac{1-\tan x}{1+\tan x}\right)\right] d x=\int_0^{\pi / 4}\) \(\log \left(\frac{2}{1+\tan x}\right) d x \)
\( =\int_0^{\pi / 4}(\log 2) d x-\int_0^{\pi / 4} \log (1+\tan x) d x=\) \(\int_0^{\pi / 4}(\log 2) d x-I \)
\( \therefore 2 I=(\log 2)[x]_0^{\pi / 4}=(\log 2)\left(\frac{\pi}{4}\right) \Rightarrow I=\left(\frac{\pi}{8}\right) \log 2\)
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
- Five students are to be arranged on a platform such that the boy \(B_1\) occupies the second position and such that the girl \(\mathrm{G}_1\) is always adjacent to the girl \(G_2\). Then, the number of such possible arrangements isMHT CET 2023 Easy
- Which of the following statement pattern is a contradiction?
\(\mathrm{S}_{1} \equiv(\mathrm{p} \rightarrow \mathrm{q}) \wedge(\mathrm{p} \wedge \sim \mathrm{q}) \ \mathrm{S}_{2} \equiv[\mathrm{p} \wedge(\mathrm{p} \rightarrow \mathrm{q})] \rightarrow \mathrm{q} \ \mathrm{S}_{3} \equiv\) \((\mathrm{p} \vee \mathrm{q}) \rightarrow \sim \mathrm{p} \ \mathrm{S}_{4} \equiv[\mathrm{p} \wedge(\mathrm{p} \rightarrow \mathrm{q})] \leftrightarrow \mathrm{q}\)MHT CET 2020 Medium - The co-ordinates of the foot of the perpendicular from the point \((1,2)\) on the line \(x-3 y+7=0\) areMHT CET 2022 Easy
- The logical statement \(\sim(p \vee q) \vee(\sim p \wedge q)\) is equivalent toMHT CET 2022 Easy
- If two dice are thrown together. Then, the probability that the sum of the numbers appearing on them is a prime number, isMHT CET 2010 Medium
- If the planes \(\bar{r} \cdot(2 \hat{i}-\lambda \hat{j}+\hat{k})=3\) and \(\bar{r} \cdot(4 \hat{i}-\hat{j}+\mu \hat{k})=5\) are parallel, then \(\lambda+\mu=\)MHT CET 2025 Easy
More PYQs from MHT CET
- What is the number of nucleons present in an atom having 29 electrons and 34 neutrons in it?MHT CET 2022 Easy
- A weak monobasic acid dissociates to \(0.001 \%\) in its \(0.01 \mathrm{M}\) solution. What is its dissociation constant?MHT CET 2022 Easy
- If \(\vec{a}=\hat{i}+\hat{j}+2 \hat{k}\) and \(\vec{b}=3 \hat{i}+2 \hat{j}-\hat{k}\), the magnitude of \([(\vec{a}+3 \vec{b}) \cdot(2 \vec{a}-\vec{b})]\) isMHT CET 2025 Medium
- An element crystallises in a fcc lattice with cell edge \(250 \mathrm{pm}\). Calculate the density of an element
(at.mass \(=90 \cdot 3\) )MHT CET 2020 Easy - Which from the following statements about \(\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_6\right]^{3+}\) complex is NOT correct?MHT CET 2024 Hard
- Two stars ' \(\mathrm{P}\) ' and ' \(\mathrm{Q}\) ' emit yellow and blue light respectively. The relation between their temperatures \(\left(\mathrm{T}_{\mathrm{P}}\right.\) and \(\left.\mathrm{T}_{\mathrm{Q}}\right)\) isMHT CET 2021 Easy