Download App

Indefinite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following integrals:
$\int\sin^5\text{x}\text{ dx}$
✓

Answer

Let $\text{I}=\int\sin^5\text{x}\text{ dx}$ Then
$\text{I}=\int\sin^3\text{x }\sin^2\text{x}\text{ dx}$
$=\int\sin^3\text{x}(1-\cos^2\text{x})\text{ dx} $
$=\int\big(\sin^3\text{x}-\sin^3\text{x}\cos^2\text{x}\big)\text{ dx}$
$=\int\big[\sin\text{x}(1-\cos^2\text{x})-\sin^3\text{x}\cos^2\text{x}\big]\text{dx}$
$=\int\big(\sin\text{x}-\sin\text{x}\cos^2\text{x}-\sin^3\text{x}\cos^2\text{x}\big)\text{dx}$
$\text{I}=\int\sin\text{x}\text{ dx}-\int\sin\text{x}\cos^2\text{x}\text{ dx}-\int\sin^3\text{x}\cos^2\text{x}\text{dx}$
Putting $\cos\text{x}=\text{t}$ and $-\sin\text{x}\text{ dx}=\text{dt}$ is $2^{nd}$ and $3^{rd}$​​​​​​​ integrals, we get
$\text{I}=\int\sin\text{x}\text{ dx}-\int\text{t}^2(-\text{dt})+\int\sin^2\text{x}\text{t}^2\text{ dt}$
$=\int\sin\text{x}\text{ dx}+\int\text{t}^2\text{dt}+\int\big(1-\cos^2\text{x}\big)\text{t}^2\text{dt}$
$=\int\sin\text{x}\text{ dx}+\int\text{t}^2\text{dt}+\int(1-\text{t}^2)\text{t}^2\text{dt}$
$=-\cos\text{x}+\frac{\text{t}^3}{3}+\frac{\text{t}^3}{3}-\frac{\text{t}^5}{5}+\text{C}$
$=-\cos\text{x}+\frac{2}{3}\text{t}^3-\frac{1}{5}\text{t}^5+\text{C}$
$=-\cos\text{x}+\frac{2}{3}(\cos^3\text{x})-\frac{1}{5}\big(\cos^5\text{x})+\text{C}$
$\therefore\ \text{I}=-\big[\cos\text{x}-\frac{2}{3}\cos^3\text{x}+\frac{1}{5}\cos^5\text{x}\big]+\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}2&3&7\\13&17&5\\15&20&12 \end{vmatrix}$
→
If the lines $\frac{\text{x}-1}{-3}=\frac{\text{y}-2}{-2\text{k}}=\frac{\text{z}-3}{2}$ and $\frac{\text{x}-1}{\text{k}}=\frac{\text{y}-2}{1}=\frac{\text{z}-3}{5}$ are perpendicular, find the value of k and, hence find the equation of the plane containing these lines.
→
In the following, find the value of the constant k so that the given function is continuous at the indicated point:
$\text{f(x)}=\begin{cases}\frac{\text{x}^2+\text{x}^2-16\text{x}+20}{(\text{x}-2)^2},&\text{ x}\neq2\\\text{k},&\text{x}=2\end{cases}$
→
Solve the following determinant equations:
$\begin{vmatrix}1&1&\text{x}\\\text{p}+1&\text{p}+1&\text{p}+\text{x}\\3&\text{x}+1&\text{x}+2\end{vmatrix}=0$
→
Evaluate the following integrals:$\int\frac{(3\sin\text{x}-2)\cos\text{x}}{13-\cos^2\text{x}-7\sin\text{x}}\text{ dx}$
→
Let S be the set of all rational numbers except 1 and * be defined on S by a * b = a + b - ab, for all a, b ∈ S.
Prove that:
  1. * is a binary operation on S.
  2. * is commutative as well as associative.
→
Solve the following differential equation:
$\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=(\text{x}-1)\text{e}^{\text{x}}$
→
A pair of fair dice is thrown. Let X be the random variable which denotes the minimum of the two numbers which appear. Find the probability distribution, mean and variance of X.
→
Evaluate the following definite integrals:$\int_{-\frac{\pi}{4}}^\limits{\frac{\pi}{4}}\frac{1}{1+\sin\text{x}}\text{ dx}$
→
In a class, $5 \%$ of the boys and $10 \%$ of the girls have an IQ of more than $150$. In this class, $60 \%$ of the students are boys. If a student is selected at random and found to have an IQ of more than $150$, find the probability that the student is a boy.
→