Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS4 Marks
Question
Evaluate the following integrals:
$\int\text{e}^{2\text{x}}\Big(\frac{1-\sin2\text{x}}{1-\cos2\text{x}}\Big)\text{dx}$
✓

Answer

WE have,
$\text{I}=\int\text{e}^{2\text{x}}\Big(\frac{1-\sin2\text{x}}{1-\cos2\text{x}}\Big)\text{dx}$
$=\int\text{e}^{2\text{x}}\Big(\frac{1-2\sin\text{x}\cos\text{x}}{2\sin^2\text{x}}\Big)\text{dx}$
Put $\text{t}=2\text{x}.$ Then $\text{dt}=2\text{dx}$
Therefore,
$\text{I}=\frac{1}{2}\int\text{e}^{\text{t}}\bigg(\frac{1-2\sin\frac{\text{t}}{2}\cos\frac{\text{t}}{2}}{2\sin^2\frac{\text{t}}{2}}\bigg)\text{dt}$
$=\frac{1}{4}\int\text{e}^{\text{t}}\bigg(\frac{1-2\sin\frac{\text{t}}{2}\cos\frac{\text{t}}{2}}{\sin^2\frac{\text{t}}{2}}\bigg)\text{dt}$
$=\frac{1}4{\int\text{e}^{\text{t}}}\bigg(\frac{1}{\sin^2\frac{\text{t}}{2}}-\frac{2\sin\frac{\text{t}}{2}\cos\frac{\text{t}}{2}}{\sin^2\frac{\text{t}}{2}}\bigg)\text{dt}$
$=\frac{1}{4}\int\text{e}^{\text{t}}\big(\text{cosec}^2\frac{\text{t}}{2}-2\cot\frac{\text{t}}{2}\big)\text{dt}$
$=-\frac{1}{4}\int\text{e}^{\text{t}}\big(2\cot\frac{\text{t}}{2}-\text{cosec}^2\frac{\text{t}}{2}\big)\text{dt}$
Consider, $\text{f(x)}=2\cot\frac{\text{t}}{2},$ then $\text{f}'\text{(x)}=-\text{cosec}^2\frac{\text{t}}{2}$
Thus, the given integrand is of the from $\text{e}^{\text{x}}\big[\text{f(x)}+\text{f'}\text{(x)}\big].$
Therefore, $\text{I}=-\frac{1}{4}\big(2\cot\frac{\text{t}}{2}\big)\text{e}^{\text{t}}+\text{C}$
$=-\frac{1}{4}\big(2\cot\frac{2\text{x}}{2}\big)\text{e}^{2\text{x}}+\text{C}$
Hence, $\int\text{e}^{2\text{x}}\Big(\frac{1-\sin2\text{x}}{1-\cos2\text{x}}\Big)\text{dx}=-\frac{1}{2}(\cot\text{x})\text{e}^{2\text{x}}+\text{C}$

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 "4 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

Find the equation of the plane through the point $2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}$ and passing throught the line of intersection of the planes $\vec{\text{r}}\cdot(\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}})=0$ and $\vec{\text{r}}\cdot(\hat{\text{j}}+2\hat{\text{k}})=0.$ 
→
$\text{If A} = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}, $ verify that $\text{A}^{2} - \text{4A - 5I = 0}$
→
If a unit vector $\vec a$ makes angles $\frac{\pi}{3}$ with $\hat i$, $\frac{\pi}{4}$ with $\hat j$ and an acute angle $\theta$ with $\hat k$, then find $\theta$ and hence, the components of $\vec a $.
→
If  xy = ex –y, show that  $\frac{\text{dy}}{\text{dx}}=\frac{\text{log x}}{\left\{\text{log(x e)}\right\}^{2}}.$
→
Prove that:
$\begin{vmatrix}\text{a}&\text{a}+\text{b}&\text{a}+2\text{b} \\\text{a}+2\text{b}&\text{a}&\text{a}+\text{b}\\\text{a}+\text{b}&\text{a}+2\text{b}&\text{a} \end{vmatrix}=9(\text{a}+\text{b})\text{b}^2$
→
Find the area enclosed between the parabola y2 = 4ax and the line y = mx.
→
Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\frac{6\text{x}}{\pi}-4\sin^{2}\text{x}\text{ on }\Big[0,\frac{\pi}{6}\Big]$
→
Evaluate the following definite integrals:
$\int\limits_{1}^{2}\frac{\text{x}}{(\text{x}+1)(\text{x}+2)}\text{ dx}$
→
Evaluate :
$
\int_0^{\frac{\pi}{2}} \frac{x \sin x \cos x}{\sin ^4 x+\cos ^4 x} d x
$
→
Evaluate the following intregals:
$\int\frac{\text{x}}{(\text{x}^2+1)(\text{x}-1)}\ \text{dx}$
→