Download App

Indefinite Integrals — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIndefinite Integrals5 Marks
Question
Evaluate the following integrals:
$\int\text{e}^{2\text{x}}\sin(3\text{x}+1)\text{dx}$
✓

Answer

We have,
$\text{I}=\int\text{e}^{2\text{x}}\sin(3\text{x}+1)\text{dx}$
Let the first function be $\sin(3x + 1)$ and the second function be $e^{2x}.$
First we find the integral of the second function, i. e., $\int\text{e}^{2\text{x}}\text{dx}$.
$\int\text{e}^{2\text{x}}\text{dx}=\frac{1}{2}\text{e}^{2\text{x}}$
Now, using integration by parts, we get
$\text{I}=\sin(3\text{x}+1)\int\text{e}^{2\text{x}}\text{dx}-\int\Big[\Big(\frac{\text{d}(\sin(3\text{x}+1))}{\text{dx}}\Big)\Big]\text{dx}$
$=\frac{1}{2}\sin(3\text{x}+1)\text{e}^{2\text{x}}\text{dx}-\frac{3}{2}\int\big[\cos(3\text{x}+1)\text{e}^{2\text{x}}\big]\text{dx}$
$=\frac{1}{2}\sin(3\text{x}+1)\text{e}^{2\text{x}}-\frac{3}{2}\Big\{\cos(3\text{x}+1)\int\text{e}^{2\text{x}}\text{dx}-\int\Big[\Big(\frac{\text{d}(\cos(3\text{x}+1))}{\text{dx}}\Big)\int\text{e}^{2\text{x}}\text{dx}\Big]\text{dx}\Big\}$
$=\frac{1}{2}\sin(3\text{x}+1)\text{e}^{2\text{x}}-\frac{3}{4}\cos(3\text{x}+1)\text{e}^{2\text{x}}-\frac{9}{4}\text{I}+\text{C}$
$\text{I}+\frac{9}{4}\text{I}=\frac{1}{2}\sin(3\text{x}+1)\text{e}^{2\text{x}}-\frac{3}{4}\cos(3\text{x}+1)\text{e}^{2\text{x}}+\text{C}$
$\frac{13}{4}\text{I}=\frac{\text{e}^{2\text{x}}}{2}\big[\sin(3\text{x}+1)-\frac{3}{2}\cos(3\text{x}+1)\big]+\text{C}$
$\text{I}=\frac{2}{13}\text{e}^{2\text{x}}\big[\sin(3\text{x}+1)-\frac{3}{2}\cos(3\text{x}+1)\big]+\text{C}$
$=\frac{\text{e}^{2\text{x}}}{13}\big[2\sin(3\text{x}+1)-3\cos(3\text{x}+1)\big]+\text{C}$
Hence, 
$\int\text{e}^{2\text{x}}\sin(3\text{x}+1)\text{dx}=\frac{\text{e}^{2\text{x}}}{13}\big[2\sin(3\text{x}+1)-3\cos(3\text{x}+1)\big]+\text{C}$

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 "5 Marks Questions" from Indefinite Integrals→Indefinite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Discuss the continuity of the following functions at the indicated point:
$\text{f}\text{(x)}=\begin{cases}\frac{{1}-\text{x}^\text{n}}{1-\text{x}}, & \text{x} \neq1\\\text{n}-1, & \text{ x} = 1\end{cases}\text{ n }\in\ \text{N at x}=1$
→
Find $x, y, z$ if $\text{A}=\begin{bmatrix}0&2\text{y}&\text{z}\\\text{x}&\text{y}&-\text{z}\\\text{x}&-\text{y}&\text{z}\end{bmatrix}$ satisfies $A′ = A^{-1}.$
→
Solve the following differential equation:
$\text{y dx}+\Big\{\text{x}\log\Big(\frac{\text{y}}{\text{x}}\Big)\Big\}\text{dy}-2\text{x dy}=0$
→
Verify Rolle's theorem for the following function on the indicated intervals
$f(x) = x^2 - 8x + 12$ on $[2, 6]$
→
If $f(x) = Ax^2 + Bx + C$ is such that $f(a) = f(b),$ then write the value of $c$ in Rolle's theorem.
→
Let A = R – {3} and B = R – {1}. Consider the function f: A → B defined by $f(\text{x})=\Big(\frac{\text{x}-2}{\text{x}-3}\Big).$ Is f one-one and onto? Justify your answer.
→
In a multiple-choice examination with three possible answers for each of the five questions out of which only one is correct, what is the probability that a candidate would get four or more correct answers just by guessing?
→
Maximum Z = 3x + 4y Subject to$\text{x}+\text{y}\leq30000$
$\text{y}\leq12000$
$\text{x}\geq6000$
$\text{x}\geq\text{y}$
$\text{x},\text{y}\geq0$
→
Find inverse, by elementary row operations (if possible), of the following matrices.
$\begin{bmatrix}1&3\\-5&7\end{bmatrix}.$
→
Find the vector equation of the plane passing through the points (1, 1, -1), (6, 4, -5) and (-4, -2, 3).
→