Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS2 Marks
Question
Evaluate the following integrals:
$\int\limits^\frac{\pi}{2}_{-\frac{\pi}{2}}\log\Big(\frac{\text{a}-\sin\theta}{\text{a}+\sin\theta}\Big)\text{d}\theta$
✓

Answer

Let $\text{I}=\int\limits^\frac{\pi}{2}_{-\frac{\pi}{2}}\log\Big(\frac{\text{a}-\sin\theta}{\text{a}+\sin\theta}\Big)\text{d}\theta$
Here, $\text{f}(\theta)=\log\Big(\frac{\text{a}-\sin\theta}{\text{a}+\sin\theta}\Big)$
Consider, $\text{f}(-\theta)=\log\bigg[\frac{\text{a}-\sin(-\theta)}{\text{a}+\sin(-\theta)}\bigg]$
 $=-\log\Big(\frac{\text{a}-\sin\theta}{\text{a}+\sin\theta}\Big)=-\text{f}(\theta)$
i.e., $\text{f}(\theta)$ is odd function.
Therefore, $\text{I}=0$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "2 Marks" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

A, B, C are three non-null square matrices of the same order, write the condition on A such that AB = AC ⇒ B = C.
→
Given: $3\left[\begin{array}{cc}x & y \\ z & w\end{array}\right]=\left[\begin{array}{cc}x & 6 \\ -1 & 2 w\end{array}\right]+\left[\begin{array}{cc}4 & x+y \\ z+w & 3\end{array}\right]$, find the values of $\mathrm{x}, \mathrm{y}, \mathrm{z}$ and $\mathrm{w}$.
→
Write the cartesian and vector equations of x-axis.
→
The mean of a binomial distribution is 10 and its standard deviation is 2; write the value of q.
→
Find the value of a, b, c and d from the following equations:
$\begin{bmatrix}2\text{a}+\text{b}&\text{a}-2\text{b}\\5\text{c}-\text{d}&4\text{c}+3\text{d}\end{bmatrix}=\begin{bmatrix}4&-3\\11&24\end{bmatrix}$
→
At what points in the interval [0, 2$\pi$], does the function sin 2x attain its maximum value?
→
Let * be a binary operation on Z defined by a * b = a + b - 4 for all a, b ∈ Z.
Find the identity element in Z.
→
Find the unit vector in the direction of $3\hat{\text{i}}+4\hat{\text{j}}-12\hat{\text{k}}$.
→
In answering a question on a multiple choice test, a student either knows the answer or guesses. Let $\frac{3}{4}$ be the probability that he knows the answer and $\frac {1}{4}$ be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability $\frac{1}{4}$. What is the probability that the student knows the answer given that he answered it correctly?
→
Prove that the following function are increasing on R.
f(x) = 4x3 + 18x2 + 27x - 27
→