Download App

Definite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals5 Marks
Question
Evaluate the following integrals:$\int\limits^{{\pi}}_{{-\pi}}\frac{2\text{x}(1+\sin\text{x})}{1+\cos^2\text{x}}\text{ dx}$
✓

Answer

Let $\text{I}=\int\limits^{{\pi}}_{{-\pi}}\frac{2\text{x}(1+\sin\text{x})}{1+\cos^2\text{x}}\text{ dx}$ Then,$\text{I}=\int\limits^{{\pi}}_{{-\pi}}\frac{2\text{x}}{1+\cos^2\text{x}}\text{ dx}+\int\limits^{{\pi}}_{{-\pi}}\frac{2\text{x}\sin\text{x}}{1+\cos^2\text{x}}\text{ dx}$
$=\text{I}_1+\text{I}_2$
Consider $\text{f(x)}=\frac{2\text{x}}{1+\cos^2-\text{x}}$ Now,$\text{f}(-\text{x})=\frac{2(-\text{x})}{1+\cos^2(\pi-\text{x})}=-\frac{2\text{x}}{1+(-\cos\text{x})^2}=-\frac{2\text{x}}{1+\cos^2\text{x}}=-\text{f(x)}$
$\therefore\ \text{I}_1=\int\limits^{{\pi}}_{{-\pi}}\frac{2\text{x}}{1+\cos^2\text{x}}\text{ dx}=0$ $\begin{bmatrix}\int\limits^{\text{a}}_{-\text{a}}\text{f(x)}\text{dx}=\begin{cases}2\int\limits^{\text{a}}_{0}\text{f(x)}\text{dx},&\text{ if }\text{f}(-\text{x})=\text{f(x)}\\0,&\text{ if }\text{f}(-\text{x})=-\text{f}(\text{x})\end{cases}\end{bmatrix}$
Again, consider $\text{g(x)}=\frac{2\text{x}\sin\text{x}}{1+\cos^2\text{x}}$$\text{g}(-\text{x})=\frac{2(-\text{x})\sin(-\text{x})}{1+\cos^2(-\text{x})}=\frac{2\text{x}\sin\text{x}}{1+\cos^2\text{x}}=\text{g(x)}$ $\big[\sin(-\text{x})=-\sin\text{x}\text{ and }\cos(-\text{x})=\cos\text{x}\big]$
$\therefore\ \text{I}_2=\int\limits^{\pi}_{-\pi}\frac{2\text{x}\sin\text{x}}{1+\cos^2\text{x}}\text{ dx}$
$=2\times2\int\limits^{\pi}_0\frac{\text{x}\sin\text{x}}{1+\cos^2\text{x}}\text{ dx}$ $\begin{bmatrix}\int\limits^{\text{a}}_{-\text{a}}\text{f(x)}\text{dx}=\begin{cases}2\int\limits^{\text{a}}_{0}\text{f(x)}\text{dx},&\text{ if }\text{f}(-\text{x})=\text{f(x)}\\0,&\text{ if }\text{f}(-\text{x})=-\text{f}(\text{x})\end{cases}\end{bmatrix}$
$=4\int\limits^{\pi}_0\frac{\text{x}\sin\text{x}}{1+\cos^2\text{x}}\text{ dx}\ ...(\text{i})$
Then,$\text{I}_2=4\int\limits^{\pi}_0\frac{(\pi-\text{x})\sin(\pi-\text{x})}{1+\cos^2(\pi-\text{x})}\text{ dx}$
$=4\int\limits^{\pi}_0\frac{(\pi-\text{x})\sin\text{x}}{1+\cos^2\text{x}}\text{ dx}\ ....(\text{ii})$
Adding (i) and (ii) we get$2\text{I}_2=4\int\limits^{\pi}_0\frac{\pi\sin\text{x}}{1+\cos^2\text{x}}\text{ dx}$
$\Rightarrow2\text{I}_2=4\int\limits^{\pi}_0\frac{\sin\text{x}}{1+\cos^2\text{x}}\text{ dx}$
Put $\cos\text{x}=\text{z}$$\Rightarrow-\sin\text{x}\text{dx}=\text{dz}$
When $\text{x}\rightarrow0,\text{ z}\rightarrow1$ When $\text{x}\rightarrow\pi,\text{ z}\rightarrow-1$$\therefore\ 2\text{I}_2=-4\pi\int\limits^{-1}_1\frac{\text{dz}}{1+\text{z}^2}$
$\Rightarrow2\text{I}_2=-4\pi\times\Big[\tan^{-1}\text{z}\Big]^{-1}_1$
$\Rightarrow2\text{I}_2=-4\pi\Big[\tan^{-1}(-1)-\tan^{-1}1\Big]$
$\Rightarrow2\text{I}_2=-4\pi\Big(-\frac{\pi}{4}-\frac{\pi}{4}\Big)=2\pi^2$
$\Rightarrow\text{I}_2=\pi^2$
$\therefore\ \text{I}=\text{I}_1+\text{I}_2$
$\Rightarrow\text{I}=0+\pi^2=\pi^2$

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 "Solve the Following Question.(5 Marks)" from Definite Integrals→Definite Integrals — 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 points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}2\text{x},&\text{ if}\text{ x}<0\\0,&\text{if }0\leq\text{x}\leq1\\4\text{x},&\text{if }\text{ x}>1\end{cases}$
→
Evaluate the following integrals:$\int\frac{\text{x}-1}{3\text{x}^2-4\text{x}+3}\text{dx}$
→
Find the slopes of the tangent and the normal to the following curves at the indicated points:
$\text{x}=\text{a}(\theta-\sin\theta),\text{y}=\text{a}(1-\cos\theta)\ \text{at}\ \theta=\frac{-\pi}{2}$
→
Three persons $A, B$ and $C$ apply for a job of Manager in a Private Company. Chances of their selection ($A, B$ and $C$) are in the ratio $1 : 2 : 4$. The probabilities that $A, B$ and $C$ can introduce changes to improve profits of the company are $0.8, 0.5$ and $0.3$, respectively. If the change does not take place, find the probability that it is due to the appointment of $C$.
→
$\int(2\text{x}^2+3)\sqrt{\text{x+2}}\text{dx}$
→
Find the general solution of the differential equation $\frac{\text{dy}}{\text{dx}}-{\text{y}}=\cos\text{x}$
→
Find the points of discontinuity, if any of the following function:
$\text{f(x)}=​​\begin{cases}-2,&\text{if }\text{ x}\leq-1\\2\text{x},&\text{if }-1<\text{x}\leq1\\2,&\text{if }\text{ x}>1\end{cases}$
→
Find the intervals in which the following functions are increasing or decreasing.
$\text{f}(\text{x})=\frac{3}{10}\text{x}^4-\frac{4}{5}\text{x}^3-3\text{x}^2+\frac{36}{5}\text{x}+11$
→
Find the relationship between 'a' and 'b' so that the function 'f' defind by $\text{f(x)}=\begin{cases}\text{ax}+1,&\text{if }\text{ x}\leq3\\\text{bx}+3,&\text{if }\text{ x}>3\end{cases}$ is continuous at x = 3.
→
Evaluate the following integrals:
$\int\frac{\text{x}^2}{(\text{a}-\text{x}^2)^{\frac{3}{2}}}\text{ dx}$
→