Download App

INTEGRALS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSINTEGRALS5 Marks
Question
$\int\text{e}^{-3\text{x}}\cos^3\text{x dx}$
✓

Answer

Let $\text{I}=\int\text{e}^{-3\text{x}}\cos^3\text{x dx}$
$=\int\text{e}^{-3\text{x}}\Big(\frac{\cos3\text{x}+3\cos\text{x}}{4}\Big)\text{dx}$
$=\frac{1}{4}\int\big(\text{e}^{-3\text{x}}\cos3\text{x}+\text{e}^{-3\text{x}}\cos\text{x}\big)\text{dx}$
$=\frac{1}{4}(\text{I}_1+\text{I}_2)$
$\text{I}_1=\int\text{e}^{-3\text{x}}\cos3\text{x dx}$
$=\text{e}^{-3\text{x}}\int\cos3\text{x dx}-\int\big((\text{e}^{-3\text{x}})\int\cos3\text{x dx}\big)\text{dx}$
$=\text{e}^{-3\text{x}}\frac{\sin3\text{x}}{3}-\int-3\text{e}^{-3\text{x}}\frac{\sin3\text{x}}{3}\text{dx}$
$=\text{e}^{-3\text{x}}\frac{\sin3\text{x}}{3}+\int\text{e}^{-3\text{x}}\sin3\text{x dx}$
$=\text{e}^{-3\text{x}}\frac{\sin3\text{x}}{3}+\text{e}^{-3\text{x}}\int\sin3\text{x dx}-\int\big((\text{e}^{-3\text{x}})\int\sin3\text{x dx})\text{dx}$
$=\text{e}^{-3\text{x}}\frac{\sin3\text{x}}{3}-\text{e}^{-3\text{x}}\frac{\cos3\text{x}}{3}-\int\text{e}^{-3\text{x}}\cos3\text{x dx}$
$=\text{e}^{-3\text{x}}\frac{3\sin}{3}-\text{e}^{-3\text{x}}\frac{\cos3\text{x}}{3}-\text{I}_1$
$\Rightarrow\ 2\text{I}_1=\frac{\text{e}^{-3\text{x}}}{3}(\sin3\text{x}-\cos3\text{x})$
$\Rightarrow\ \text{I}_1=\frac{\text{e}^{-3\text{x}}}{6}(\sin3\text{x}-\cos3\text{x})+\text{C}_1$
Similarly $\text{I}_2=\int\text{e}^{-3\text{x}}\cos\text{x dx}=\frac{\text{e}^{-3\text{x}}}{10}(\sin3\text{x}-3\cos3\text{x})+\text{C}_2$
$\Rightarrow\ \text{I}=\frac{1}{4}\bigg[\frac{\text{e}^{-3\text{x}}}{6}(\sin3\text{x}-\cos3\text{x})+\frac{\text{e}^{-3\text{x}}}{10}(\sin3\text{x}-3\cos3\text{x})\bigg]+\text{C}$

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 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:
$\int\limits^2_{-2}\text{xe}^{|\text{x}|}\text{ dx}$ 
→
If $\text{x}=\text{a}(\cos\theta+\theta\sin\theta),\text{y}=\text{a}(\sin\theta-\theta\cos\theta)$ prove that $\frac{\text{d}^2\text{x}}{\text{d}\theta^2}=\text{a}(\cos\theta-\theta\sin\theta),\frac{\text{d}^2}{\text{d}\theta^2}$ $=\text{a}(\sin\theta-\theta\cos\theta)\ \text{and}\ \frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\sec^3\theta}{\text{a}\theta}$
→
Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}-\text{x}\log\text{x}$
→
Find the probability distribution of the number of doublets in three throws of a pair of dice and find its mean.
→
An article manufactured by a company consists of two parts X and Y. In the process of manufacture of the part X, 9 out of 100 parts may be defective. Similarly, 5 out of 100 are likely to be defective in the manufacture of part Y. Calculate the probability that the assembled product will not be defective.
→
Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\frac{\sin\text{x}}{\text{e}^{\text{x}}}\text{ on }0\leq\text{x}\leq\pi$
→
$A$ and $B$ throw a pair of dice alternately. $A $wins the game if he gets a total of $7$ and $B$ wins the game if he gets a total of $10.$ If $A$ starts the game, then find the probability that $B$ wins.
→
If the mean and variance of a random variable X having  a binomial distribution are 4 and 2 respectively, find P (X = 1).
→
A(1, 0, 4) B(0, -11, 1), C(2, -3, 1) are three points and D is the fool of perpendicular from A on BC. Find the coordinates of D.
→
Using the method of integration find the area bounded by the curve |x| + |y| = 1
[Hint: the required region is bounded by lines x + y = 1, x - y = 1, -x + y = 1 and -x - y = 11]
→