Download App

Integrals — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIntegrals5 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 "5 Marks Questions" from Integrals→Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

If x, y, z are different and $\begin{vmatrix} \text{x} & \text{x}^{2} & \text{1 + x}^{3} \\ \text{y} & \text{y}^{2} & \text{1 + y}^{3} \\ \text{z} & \text{z}^{2} & \text{1 + z}^{3} \end{vmatrix}$= 0, show that xyz = -1.
→
Discuss the continuity and differentiability of $\text{f(x)}=\text{e}^{|\text{x}|}.$
→
Solve the following equation for x:
$\tan^{-1}\frac{1}{4}+2\tan^{-1}\frac{1}{5}+\tan^{-1}\frac{1}{6}+\tan^{-1}\frac{1}{\text{x}}=\frac{\pi}{4}$
→
Find the adjoint of the matrix $\text{A}=\begin{bmatrix} -1 & -2 & -2 \\ 2 & 1 & -2 \\ 2 & -2 & 1 \end{bmatrix}$ and hence show that $\ce{A (adj A) = |A|I_3.}$
→
One kind of cake requires $300\ gm$ of flour and $15\ gm$ of fat, another kind of cake requires $150\ gm$ of flour and $30\ gm$ of fat. Find the maximum number of cakes which can be made from $7.5\ kg$ of flour and $600\ gm$ of fat, assuming that there is no shortage of the other ingradients used in making the cake. Make it as an $LPP$ and solve it graphically.
→
Form the differential equation of the family of curves represented by the equation (a being the perimeter):$(2\text{x}+\text{a})^2+\text{y}^2=\text{a}^2$
→
Evaluate the following integrals:
$\int\text{x}\cos^3\text{x}^2\sin\text{x}^2\text{ dx}$
→
Solve the system of linear equation, using matrix method $x - y + 2z = 7; 3x + 4y - 5z = -5; 2x - y + 3z = 12$
→
If $\text{A}=\begin{bmatrix}2&3\\1&2\end{bmatrix}$ and $\text{I}=\begin{bmatrix}1&0\\0&1\end{bmatrix},$ then find $\lambda,\mu$ so that $\text{A}^2=\lambda\text{A}+\mu\text{I}$
→
If $\text{y}=(\tan\text{x})^{(\tan\text{x})^{(\tan\text{x})^{....\infty}}},$ prove that $\frac{\text{dy}}{\text{dx}}=2\text{ at x}=\frac{\pi}{4}$
→