Download App

Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
Question
Evaluate the definite integral $\int_{-\frac{\pi}{2}}^{\pi} e^{x}\left(\frac{1-\sin x}{1-\cos x}\right) d x$
✓

Answer

Let $I=\int_{-\frac{\pi}{2}}^{\pi}\left(e^{x}\left(\frac{1-\sin x}{1-\cos x}\right) d x\right.$ 
= $\int_{-\frac{\pi}{2}}^{\pi}\left(e^{x}\left(\frac{1-2 \sin \frac{x}{2} \cos \frac{x}{2}}{2 \sin ^{2}\left(\frac{x}{2}\right)}\right) d x\right.$ 
= $\int_{-\frac{\pi}{2}}^{\pi}\left(e^{x}\left(\frac{1}{2 \sin ^{2}\left(\frac{x}{2}\right)}-\frac{2 \sin \frac{x}{2} \cos \frac{x}{2}}{2 \sin ^{2}\left(\frac{x}{2}\right)}\right) d x\right.$ 
= $\int_{-\frac{\pi}{2}}^{\pi}\left(e^{x}\left(\frac{1}{2} \csc ^{2}\left(\frac{x}{2}\right)-\cot \frac{x}{2}\right) d x\right.$ 
Now let $f(x)=-\cot \frac{x}{2}$ 
$\Rightarrow$ f'(x) = $-\left(-\frac{1}{2} \csc ^{2}\left(\frac{x}{2}\right)\right)=\frac{1}{2} \csc ^{2}\left(\frac{x}{2}\right)$ 
$\Rightarrow \int_{-\frac{\pi}{2}}^{\pi}\left(e^{x}\left(\frac{1}{2} \csc ^{2}\left(\frac{x}{2}\right)-\cot \frac{x}{2}\right) d x\right.$ = $\int_{-\frac{\pi}{2}}^{\pi}\left(f(x)+f^{\prime}(x)\right) e^{x} d x$ 
= $\left[e^{x} f(x)\right]_{-\frac{\pi}{2}}^{\pi}$ 
= $\left[e^{x}\left(-\cot \frac{x}{2}\right)\right]_{-\frac{\pi}{2}}^{\pi}$ 
= $-\left[e^{\pi}\left(\cot \frac{\pi}{2}\right)-e^{\frac{\pi}{2}}\left(\cot \frac{\pi}{4}\right)\right]$
= $-\left[e^{\pi}(0)-e^{\frac{\pi}{2}}(1)\right]$ 
= $-\left[0-e^{\frac{\pi}{2}}\right]$ 
$\Rightarrow I=e^{\frac{\pi}{2}}$

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 "1 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

Is the measure of 20 m/s towards north is scalar or vector?
→
Write the degree of the differrntial equation $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}+3\Big(\frac{\text{dy}}{\text{dx}}\Big)^{2}=\text{x}^{2}\log\Big(\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}\Big).$ 
→
Evaluate: $\int\limits_0^{1/\sqrt{2}}\frac{1}{\sqrt{\text{1 - x}^{2}}}\text{dx}.$
→
Given an example of:
A row matrix which is also a column matrix,
→
Evaluate the following:
$\cot^{-1}\Big(\cot\frac{\pi}{3}\Big)$
→
Is the measure of 30 km/hr scalar or vector quantity?
→
A die is thrown. If E is the event the number appearing is a multiple of 3 and F be the event the number appearing is even then find whether E and F are independent?
→
Find the interval in function 6 - 9x - x2 is increasing or decreasing.
→
Write the vector equation of the plane, passing through the point (a, b, c) and parallel to the plane $\overrightarrow{\text{r}}.(\hat{\text{i}} + \hat{\text{j}} + \hat{\text{k}}) = 2.$
→
Write the value of p for which $\overrightarrow{a} = 3\hat{i} + 2\hat{j} + 9\hat{k}$ and  $\overrightarrow{b} = \hat{i} + \text{p}\hat{j} + 3\hat{k}$ are parallel vectors.
→