Download App

Definite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals5 Marks
Question
Evaluate the following definite integrals:
$\int\limits_{0}^{\pi}\text{e}^{2\text{x}}\sin\Big(\frac{\pi}{4}+{\text{x}}\Big)\text{dx}$
✓

Answer

Let $\text{I}=\int_{0}^\limits{\pi}\text{e}^{2\text{x}}\sin\Big(\frac{\pi}{4}+{\text{x}}\Big)\text{dx}$
Integrating by parts, we get
$\text{I}=\frac{1}{2}\Big[\text{e}^{2\text{x}}\sin\Big(\frac{\pi}{4}+{\text{x}}\Big)\text{dx}\Big]^\pi_0-\frac{1}{2}\int_{0}^\limits{\pi}\text{e}^{2\text{x}}\cos\Big(\frac{\pi}{4}+{\text{x}}\Big)\text{dx}$
Now, integrating the second term by parts, we get
$\Rightarrow\text{I}=\frac{1}{2}\Big[\text{e}^{2\text{x}}\sin\Big(\frac{\pi}{4}+{\text{x}}\Big)\text{dx}\Big]^\pi_0-\frac{1}{2}\bigg\{\Big[\frac{1}{2}\text{e}^{2\text{x}}\cos\Big(\frac{\pi}{4}+{\text{x}}\Big)\Big]^{\pi}_0\\+\frac{1}{2}\int_{0}^\limits{\pi}\text{e}^{2\text{x}}\sin\Big(\frac{\pi}{4}+{\text{x}}\Big)\text{dx}\bigg\}$
$\Rightarrow\text{I}=\frac{1}{2}\Big[\text{e}^{2\text{x}}\sin\Big(\frac{\pi}{4}+{\text{x}}\Big)\text{dx}\Big]^\pi_0-\frac{1}{4}\Big[\text{e}^{2\text{x}}\cos\Big(\frac{\pi}{4}+{\text{x}}\Big)\text{dx}\Big]^{\pi}_0-\frac{1}{4}\text{I}$
$\Rightarrow\frac{5}{4}\text{I}=\frac{1}{2}\Big[\text{e}^{2\pi}\sin\Big(\pi+\frac{\pi}{4}\Big)-\sin\Big(\frac{\pi}{4}\Big)\Big]\\-\frac{1}{4}\Big[\text{e}^{2\pi}\cos\Big(\pi+\frac{\pi}{4}\Big)-\cos\Big(\frac{\pi}{4}\Big)\Big]$
$\Rightarrow\frac{5}{4}\text{I}=\frac{1}{2}\Big[-\text{e}^{2\pi}\times\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}\Big]-\frac{1}{4}\Big[-\text{e}^{2\pi}\times\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}\Big]$
$\Rightarrow\frac{5}{4}\text{I}=-\frac{1}{2\sqrt{2}}\text{e}^{2\pi}-\frac{1}{2\sqrt{2}}+\frac{1}{4\sqrt{2}}\text{e}^{2\pi}+\frac{1}{4\sqrt{2}}$
$\Rightarrow\text{I}=-\frac{1}{5\sqrt{2}}\big(\text{e}^{2\pi}+1\big)$

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 "5 Marks Questions" from Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\begin{bmatrix}4-\text{x}&4+\text{x}&4+\text{x}\\4+\text{x}&4-\text{x}&4+\text{x}\\4+\text{x}&4+\text{x}&4-\text{x}\end{bmatrix}=0,$ then find values of $x$.
→
If y(x) is a solution of $\Big(\frac{2+\sin\text{x}}{1+\text{y}}\Big)\frac{\text{dy}}{\text{dx}}=-\cos\text{x}$ and y(0) = 1, then find the value of $\text{y}\big(\frac{\pi}{2}\big).$
→
Using properties of determinants, prove the following:$\begin{vmatrix} \alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta + \gamma & \gamma + \alpha & \alpha + \beta \end{vmatrix} = (\alpha - \beta)(\beta - \gamma)(\gamma - \alpha)( \alpha + \beta + \gamma) $
→
Show that the line whose vector equation is $\vec{\text{r}}=2\hat{\text{i}}+5\hat{\text{j}}+7\hat{\text{k}}+\lambda(\hat{\text{i}}+3\hat{\text{j}}+4\hat{\text{k}})$ is parallel to the plane whose vector equation is $\vec{\text{r}}\cdot(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}})=7$ Also, find the distance between thetm.
→
The two vectors $\hat{\text{j}}+\hat{\text{k}}$ and $3\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}$ represents the sides $\overrightarrow{\text{AB}}$ and $\overrightarrow{\text{AC}}$ respectively of a triangle ABC. Find the length of the median through A.
→
Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}1&43&6\\7&35&4\\3&17&2\end{vmatrix}$
→
Solve the following differential equation
$\sqrt{\text{a}+\text{x}}\text{dy}+\text{x dx}=0$
→
If $\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\ 2\hat{\text{i}}+5\hat{\text{J}},\ 3\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}$ and $\hat{\text{i}}-6\hat{\text{j}}-\hat{\text{k}}$ respectively are the position vectors of points A, B, C and D, then find the angle between the straight lines AB and CD. Find whether $\vec{\text{AB}}$ and $\vec{\text{CD}}$ are collinear or not.
→
A firm manufactures two products, each of which must be processed through two departments, 1 and 2. The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows:
 
 
Product A
Product B
Weekly capacity
Department 1
3
2
130
Department 2
4
6
260
Selling price per unit
Rs. 25
Rs. 30
 
Labour cost per unit
Rs. 16
Rs. 20
 
Raw material cost per unit
Rs. 4
Rs. 4
 
The problem is to determine the number of units to produce each product so as to maximize total contribution to profit. Formulate this as a LPP.
→
Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\sin\text{x}\sin2\text{x}\sin3\text{x}\sin4\text{x}$
→