Download App

Definite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integration4 Marks
Question
Evaluate : $\int_0^{\pi / 4} \frac{\sec ^2 x}{2 \tan ^2 x+5 \tan x+1} \cdot d x$
✓

Answer

Let $\mathrm{I}=\int_0^{\pi / 4} \frac{\sec ^2 x}{2 \tan ^2 x+5 \tan x+1} \cdot d x$
put $\tan x=t \quad \therefore \sec ^2 x \cdot d x=1 \cdot d t$
As $x$ varies from 0 to $\frac{\pi}{4}$
$t$ varies from 0 to 1
$
\begin{aligned}
& =\int_0^1 \frac{1}{2 t^2+4 t+1} \cdot d t \\
& =\frac{1}{2} \cdot \int_0^1 \frac{1}{t^2+2 t+\frac{1}{2}} \cdot d t \\
& =\frac{1}{2} \cdot \int_0^1 \frac{1}{t^2+2 t+1-1+\frac{1}{2}} \cdot d t \\
& =\frac{1}{2} \cdot \int_0^1 \frac{1}{(t+1)^2-\left(\frac{1}{\sqrt{2}}\right)^2} \cdot d t
\end{aligned}
$
$\begin{aligned} & =\frac{1}{2} \frac{1}{2\left(\frac{1}{\sqrt{2}}\right)}\left[\log \left[\frac{(t+1)-\frac{1}{\sqrt{2}}}{(t+1)+\frac{1}{\sqrt{2}}}\right]\right]_0^1 \\ & =\frac{\sqrt{2}}{4} \log \left[\left(\frac{\sqrt{2} t+\sqrt{2}-1}{\sqrt{2} t+\sqrt{2}+1}\right)\right]_0^1 \\ & =\frac{\sqrt{2}}{4}\left[\log \left(\frac{\sqrt{2}(1)+\sqrt{2}-1}{\sqrt{2}(1)+\sqrt{2}+1}\right)-\log \left(\frac{\sqrt{2}(0)+\sqrt{2}-1}{\sqrt{2}(0)+\sqrt{2}+1}\right)\right] \\ & =\frac{\sqrt{2}}{4}\left[\log \left(\frac{2 \sqrt{2}-1}{2 \sqrt{2}+1}\right)-\log \left(\frac{\sqrt{2}-1}{\sqrt{2}+1}\right)\right] \\ & =\frac{\sqrt{2}}{4} \log \left[\left(\frac{2 \sqrt{2}-1}{2 \sqrt{2}+1}\right) \div\left(\frac{\sqrt{2}-1}{\sqrt{2}+1}\right)\right] \\ & =\frac{\sqrt{2}}{4} \log \left[\frac{3+\sqrt{2}}{3-\sqrt{2}}\right]\end{aligned}$

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 "Solve the Following Question.(4 Marks)" from Definite Integration→Definite Integration — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the general solutions of the following :

$\tan ^3 \theta=3 \tan \theta$

→
Find the area enclosed between the circle $x^2 + y^2 = 1$ and the line $x + y = 1$, lying in the first quadrant
→
Find the area of the region bounded by the curve $(y − 1)^2 = 4(x + 1)$ and the line$ y = (x − 1)$
→
Find the ratio in which point $P$ divides $A B$ and $C D$ where $A(2,-3,4), B(0,5,2), C(-1,5,3)$ and $\mathrm{D}(2,-1,3)$. Also, find its coordinates.
→
The curve passes through the point (0, 2). The sum of the coordinates of any point on the curve exceeds the slope of the tangent to the curve at any point by 5. Find the equation of the curve.
→
Find the area of the region bounded by the curves. $y^2=4 x$ and $4 x^2+4 y^2=9$ with $x \geq 0$.
→
If $A=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right], B=\left[\begin{array}{ll}1 & 0 \\ 3 & 1\end{array}\right]$, find $A B$ and $(A B)^{-1}$
→
Solve the following differential equation
$y x \frac{ d y}{ d x}= x ^2+2 y ^2$
→
Find the general solution of : $\cos x-\sin x=1$.
→
Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find E(X).
→