Download App

Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals4 Marks
Question
By using the properties of definite integrals, evaluate the integral $\int_{0}^{\frac{\pi}{4}} \log (1+\tan x) d x$
✓

Answer

Given $\int_{0}^{\frac{\pi}{4}} \log (1+\tan x) d x$ 
Let $I=\int_{0}^{\frac{\pi}{4}} \log (1+\tan x) d x$ .....(i)
as, $\left\{\int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x\right\}$ 
$\Rightarrow \mathrm{I}=\int_{0}^{\frac{\pi}{4}} \log \left[1+\tan \left(\frac{\pi}{4}-\mathrm{x}\right)\right] \mathrm{d} \mathrm{x}$ 
as $\left\{\tan (A-B)=\frac{\tan (A)-\tan (B)}{1+\tan (A) \tan (B)}\right\}$ 
$\Rightarrow \mathrm{I}=\int_{0}^{\frac{\pi}{4}} \log \left[1+\frac{\tan \left(\frac{\pi}{4}\right)-\tan (\mathrm{x})}{1+\tan \left(\frac{\pi}{4}\right) \tan (\mathrm{x})}\right] \mathrm{dx}$ 
$\Rightarrow \mathrm{I}=\int_{0}^{\frac{\pi}{4}} \log \left[1+\frac{1-\tan (\mathrm{x})}{1+\tan (\mathrm{x})}\right] \mathrm{d} \mathrm{x}$ 
$\Rightarrow I=\int_{0}^{\frac{\pi}{4}} \log \left[\frac{2}{1+\tan (x)}\right] d x$ 
$\Rightarrow \mathrm{I}=\int_{0}^{\frac{\pi}{4}} \log [2] \mathrm{d} \mathrm{x}-\int_{0}^{\frac{\pi}{4}} \log [1+\tan (\mathrm{x})] \mathrm{d} \mathrm{x}$ 
$\Rightarrow \mathrm{I}=\int_{0}^{\frac{\pi}{4}} \log [2] \mathrm{d} \mathrm{x}-\mathrm{I}$ (from (i))
$\Rightarrow 2 \mathrm{I}=[\mathrm{x} \log 2]_{0}^{\frac{\pi}{4}}$ 
$\Rightarrow 2 \mathrm{I}=\frac{\pi}{4} \log 2-0$ 
$\Rightarrow \mathrm{I}=\frac{\pi}{8} \log 2$

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 "4 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

Differentiate the following functions from first principles:
$\text{e}^{\sqrt{\cot\text{x}}}$
→
Find one-parameter families of solution curves of the following differential equation: (or solve the following differential equation)

$(\text{x + y})\frac{\text{dy}}{\text{dx}}=1$

→
If $y=e^{a \cos ^{-1} x}$, show that $\left(1-x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}-a^2 y=0$, where $-1 \leq x \leq 1$
→
Evaluate the following definite integrals:
$\int\limits_{\text{e}}^{\text{e}^2}\Big\{\frac{1}{\log\text{x}}-\frac{1}{(\log\text{x})^2}\Big\}\text{dx}$
→
Solve the following initial value problems:
$\frac{\text{dy}}{\text{dx}}-\frac{\text{y}}{\text{x}}+\text{cosec}\frac{\text{y}}{\text{x}}=0,\text{y}(1)=0$
→
Evaluate: $\int$sin x sin 2x sin 3x dx.
→
Evaluate the following integrals as limit of sum:
$\int\limits^4_{1}\big(\text{x}^2-\text{x}\big)\text{dx}$
→
Write the number of points where f(x) = |x| + |x − 1| is continuous but not differentiable.
→
If the vectors $\vec{\text{a}}=2\hat{\text{i}}-3\hat{\text{j}}$ and $\vec{\text{b}}=-6\hat{\text{i}}+\text{m}\hat{\text{j}}$ are collinear, find tghe value of m.
→
Verify Rolle's theorem for the following function on the indicated intervals
f(x) = x2 - 8x + 12 on [2, 6]
→