Download App

Definite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals5 Marks
Question
Evaluate the following integrals:$\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\sin^4\text{x dx}$
✓

Answer

Let $\text{I}=\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\sin^4\text{x dx}$$=\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\big(\sin^2\text{x}\big)^2\text{dx}$
$=\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\Big(\frac{1-\cos2\text{x}}{2}\Big)^2\text{dx}$
$=\frac{1}{4}\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\big(1-2\cos2\text{x}+\cos^22\text{x}\big)\text{dx}$
$=\frac{1}{4}\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\text{dx}-\frac{1}{2}\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\cos2\text{x dx}+\frac{1}{8}\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}(1+\cos4\text{x})\text{dx}$
$=\frac{1}{4}\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\text{dx}-\frac{1}{2}\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\cos2\text{x dx}+\frac{1}{8}\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\text{dx}+\frac{1}{8}\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\cos4\text{x dx}$
$=\frac{1}{4}\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\text{dx}-\frac{1}{2}\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\cos2\text{x dx}+\frac{1}{8}\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\cos4\text{x dx}$
$=\frac{3}{8}\Big[\text{x}\Big]^{\frac{\pi}{2}}_{-\frac{\pi}{2}}-\frac{1}{4}\Big[\sin2\text{x}\Big]^{\frac{\pi}{2}}_{-\frac{\pi}{2}}+\frac{1}{32}\Big[\sin4\text{x}\Big]^{\frac{\pi}{2}}_{-\frac{\pi}{2}}$
$=\frac{3}{8}\Big(\frac{\pi}{2}+\frac{\pi}{2}\Big)-\frac{1}{4}(0-0)+\frac{1}{32}(0-0)$
Hence, $\text{I}=\frac{3\pi}{8}$

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 Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $\text{A}=\begin{bmatrix}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix},$ show that $A^{-1} = A^3.$
→
Differentiate the following functions with respect to x:
$\sin^{-1}\Big(\frac{\text{x}}{\sqrt{\text{x}^2+\text{x}^2}}\Big)$
→
Solve the following differential equation:
$\text{x}^2\frac{\text{dy}}{\text{dx}}=\text{x}^2-2\text{y}^2+\text{xy}$
→
Using the method of interation, find the area of the region bounded by the following lines:
3x - y - 3 = 0, 2x + y - 12 = 0, x - 2y - 1 = 0.
→
On Z, the set of all integers, a binary operation * is defined by a * b = a + 3b - 4. Prove that * is neither commutative nor associative on Z.
→
If $\text{A}=\begin{bmatrix}0&0\\4&0\end{bmatrix},$ find $A^{16}.$
→
Solve the following systems of linear equations by cramer's rule:
5x - 7y + z = 11,
6x - 8y - z = 15,
3x + 2y - 6z = 7
→
Solve the following differential equation
$\text{x}\cos\text{y dy}=(\text{xe}^\text{x}\log\text{x}+\text{e}^\text{x})\text{dx}$
→
Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards, find the probability that
(i) all the five cards are spades
(ii) only 3 cards are spades
(iii) none is a spade.
→
Solve the following equation for x:
$\tan^{-1}\Big(\frac{\text{x}-2}{\text{x}-1}\Big)+\tan^{-1}\Big(\frac{\text{x}+2}{\text{x}+1}\Big)=\frac{\pi}{4}$
→