Download App

INTEGRALS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSINTEGRALS3 Marks
Question
Evaluate the definite integral in Exercise:
$\int\limits_\frac{\pi}{2}^{\pi}\text{e}^{\text{x}}\bigg(\frac{1-\sin\text{x}}{1+\cos\text{x}}\bigg)\text{dx}$
✓

Answer

$\text{I}=\int^{\pi}\limits_{\frac{\pi}{2}}\text{e}^{\text{x}}\bigg(\frac{1-\sin\text{x}}{1+\cos\text{x}}\bigg)\text{dx}$ $\text{I}=\int^{\pi}\limits_{\frac{\pi}{2}}\text{e}^{\text{x}}\Bigg[\frac{1-2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}}{2\sin^{2}\frac{\text{x}}{2}}\Bigg]\text{dx}$ $\text{I}=\int^{\pi}\limits_{\frac{\pi}{2}}\text{e}^{\text{x}}\Bigg[\frac{\text{cosec}^{2}\frac{\text{x}}{2}}{2}-\cot\frac{\text{x}}{2}\Bigg]\text{dx}$ $\text{Let f (x)}=-\cot\frac{\text{x}}{2}$ $\Rightarrow\text{f (x)}=-\bigg(-\frac{1}{2}\text{cosec}^{2}\frac{\text{x}}{2}\bigg)=\frac{1}{2}\text{cosec}^{2}\frac{\text{x}}{2}$ $\therefore\ \text{I}=\int^{\pi}\limits_{\frac{\pi}{2}}\text{e}^{\text{x}}\Big[\text{f(x)}+\text{f}'\text{(x)}\Big]\text{dx}$$=\Big[\text{e}^{\text{x}}.\text{f (x) dx}\Big]_{\frac{\pi}{2}}^{\pi}$
$=-\bigg[\text{e}^\text{x}.\cot\frac{\text{x}}{2}\bigg]^{\pi}_{\frac{\pi}{2}}$ $=-\bigg[\text{e}^{\text{x}}\times\cot\frac{\pi}{2}-\text{e}^{\frac{\pi}{2}}\times\cot\frac{\pi}{4}\bigg]$ $=-\bigg[\text{e}^{\text{x}}\times0-\text{e}^{\frac{\pi}{2}}\times1\bigg]$$=\text{e}^{\frac{\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 "3 Marks Question" 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

Evaluate the following integrals:$\int\frac{\log\text{x}}{\text{x}^{\text{n}}}\text{dx}$
→
Evaluate $\sin\Big(\tan^{-1}\frac{3}{4}\Big).$
→
In the following, determine the values of constants involved in the definition so that the given function is continuous:
$\text{f(x)}=\begin{cases}\frac{\sin2\text{x}}{5\text{x}},&\text{if }\text{ x}\neq0\\3\text{k},&\text{if }\text{ x}=0\end{cases}$ 
→
Evaluate the following integrals:
$\int\big(2-3\text{x}\big)\big(3+2\text{x}\big)\big(1-2\text{x}\big)\text{dx}$
→
Show that the line joining the origin to the point $(2, 1, 1)$ is perpendicular to the line determined by the points $(3, 5, -1)$ and $(4, 3, -1).$
→
Evaluate the following integrals:
$\int\frac{\cos\text{x}-\sin\text{x}}{1+\sin2\text{x}}\text{dx}$
→
Solve the differential equation $\left(\tan ^{-1} y-x\right) d y=\left(1+y^2\right) d x$
→
Evaluate the following integrals:$\int\sqrt{\text{cosec}\text{x}-1}\text{ dx}$
→
Discuss the continuity of $\text{f}\text{(x)}=\begin{cases}2\text{x}-1, & \text{x} < 0\\2\text{x}+1, & \text{x} \geq 0\end{cases}\text{at}\text{ x}=0$ 
→
If $A^{-1}=\left[\begin{array}{ccc}3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1\end{array}\right]$ find $( A B)^{-1}$
→