Download App

Indefinite Integrals — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIndefinite Integrals5 Marks
Question
Evaluate the following integrals: $\int\cot^6\text{x}\text{ dx}$
✓

Answer

$\int\cot^6\text{x}\text{ dx}$
$=\int\cot^4\text{x}\cdot\big(\text{cosec}^2-1\big)\text{dx}$
$=\int\cot^4\text{x}\cdot\text{ cosec}^2\text{x}\text{ dx}-\int\cot^4\text{x}\text{dx}$
$=\int\cot^4\text{x}\cdot\text{ cosec}^2\text{x}\text{ dx}-\int\cot^2\text{x}\cdot\cot^2\text{x}\text{ dx}$
$=\int\cot^4\text{x}-\text{ cosec}^2\text{x}\text{ dx}-\int\big(\text{cosec}^2\text{x}-1\big)\cot^2\text{x}\text{ dx}$
$=\int\cot^4\text{x}\cdot\text{ cosec}^2\text{x}\text{ dx}-\int\cot^2\text{x}\cdot\text{cosec}^2\text{x}\text{ dx}+\int\cot^2\text{x}\text{ dx}$
$=\int\cot^4\text{x}\cdot\text{ cosec}^2\text{x}\text{ dx}-\int\cot^2\text{x}\cdot\text{cosec}^2\text{x}\text{ dx}+\int(\text{cosec}^2\text{x}-1)\text{ dx}$
Now, let $\text{I}_1=\int\cot^4\text{x}\cdot\text{cosec}^2\text{x}\text{ dx}-\int\cot^2\text{x}\cdot\text{cosec}^2\text{x}\text{ dx}$
And $\text{I}_2=\int(\text{cosec}^2\text{x}-1)\text{dx}$
First we integrate $I_1$
$\text{I}_1=\int\cot^4\text{x}\cdot\text{cosec}^2\text{x}\text{ dx}-\int\cot^2\text{x}\cdot\text{cosec}^2\text{x}\text{ dx}$
Let $\cot\text{x}=\text{t}$
$-\text{cosec}^2\text{x}\text{ dx}=\text{dt}$
$\text{cosec}^2\text{x}\text{ dx}=-\text{dt}$
$\text{I}_1=-\int\text{t}^4\text{dt}+\int\text{t}^2\text{dt}$
$=\frac{-\text{t}^5}{5}+\frac{\text{t}^3}{3}+\text{C}_1$
$=-\frac{\cot^5\text{x}}{5}+\frac{\cot^3\text{x}}{3}+\text{C}_1$
Now we integrate $I_2$
$\text{I}_2=\int(\text{cosec}^2\text{x}-1)\text{dx}$
$=-\cot\text{x}-\text{x}+\text{C}_1$
Now, $\int\cot^6\text{x}\text{ dx}=\text{I}_1+\text{I}_2$
$=-\frac{1}{5}\cot^5\text{x}+\frac{1}{3}\cot^3\text{x}-\cot\text{x}-\text{x}+\text{x}+\text{C}_1+\text{C}_2$
$=-\frac{1}{5}\cot^5\text{x}+\frac{1}{3}\cot^3\text{x}-\cot\text{x}-\text{x}+\text{C}$ $\big[\therefore\text{ C}=\text{C}_1+\text{C}_2\big]$

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 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

Find the shortest distance between the lines
$\vec{\text{r}}=\big(\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}\big)+\lambda\big(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big)$ and, $\vec{\text{r}}=2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}+\mu\big(2\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}\big)$
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{3}}_{\frac{\pi}{6}}\frac{\sqrt{\sin\text{x}}}{\sqrt{\sin\text{x}}+\sqrt{\cos\text{x}}}\text{ dx}$
→
Find the area bounded by the circle $x^2+y^2=16$ and the line $\sqrt{3} y=x$ inthe first quadrant, using integration
→
Show that the lines $\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7}$ and $\frac{x-2}{1}=\frac{y+4}{3}=\frac{z-6}{5}$ intersect each other. Find also the coordinates of the point of intersection.
→
If $\text{A}=\begin{bmatrix}\cos\theta&\text{i}\sin\theta\\\text{i}\sin\theta&\cos\theta\end{bmatrix},$ then prove by principle of mathematical induction that $\text{A}^\text{n}=\begin{bmatrix}\cos\text{n}\theta&\text{i}\sin\text{n}\theta\\\text{i}\sin\text{n}\theta&\cos\text{n}\theta\end{bmatrix}$ for all $\text{n}\in\text{N}.$
→
The contents of urns $I, II, III$ are as follows:
Urn $I : 1$ white, $2$ black and $3$ red balls
Urn $II : 2$ white, $1$ black and $1$ red balls
Urn $III : 4$ white, $5$ black and $3$ red balls.
One urn is chosen at random and two balls are drawn. They happen to be white and red. What is the probability that they come from Urns $I, II, III$?
→
Solve the following differential equation: $y^2dx + (x^2 – xy + y^2)dy = 0$
→
Differentiate the following w.r.t. x:
$\tan^{-1}\bigg(\frac{\sqrt{1+\text{x}^2}+\sqrt{1-\text{x}^2}}{\sqrt{1+\text{x}^2}-\sqrt{1-\text{x}^2}}\bigg),-1<\text{x}<1,\text{ x}\neq0$
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\frac{1}{1+\cot^{\frac{3}{2}}\text{x}}\text{ dx}$
→
Solve the following differential equation : $\frac{\text{dy}}{\text{dx}}+2\text{y}=\sin\text{x}$
→