Download App

Definite Integrals — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDefinite Integrals5 Marks
Question
Evaluate the following definite integrals:
$\int\limits_{0}^{\frac{\pi}{2}}\cos^4\text{x}\text{ dx}$
✓

Answer

We have,
$\int_{0}^\limits{\frac{\pi}{2}}\cos^4\text{x}\text{ dx}$
$=\frac{1}{4}\int_{0}^\limits{\frac{\pi}{2}}(1+\cos2\text{x})^2\text{dx}$ $\big[\because2\cos^2\text{x}=1+\cos2\text{x}\big]$
$=\frac{1}{4}\int_{0}^\limits{\frac{\pi}{2}}\big(1+\cos^22\text{x}+2\cos2\text{x}\big)\text{dx}$
$=\frac{1}{4}\int_{0}^\limits{\frac{\pi}{2}}\Big(1+\frac{1+\cos4\text{x}}{2}+2\cos2\text{x}\Big)\text{dx}$
$=\frac{1}{4}\Big[\text{x}+\frac{1}{2}\text{x}+\frac{\sin4\text{x}}{8}+\sin2\text{x}\Big]^{\frac{\pi}{2}}_0$ $\Big[\because\int\cos4\text{x dx}=\frac{\sin4\text{x}}{4}\Big]$
$=\frac{1}{4}\Big[\frac{\pi}{2}+\frac{\pi}{4}+0+0-0-0-0-0\Big]$
$=\frac{1}{4}\times\frac{3\pi}{4}$
$=\frac{3\pi}{16}$

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 Definite Integrals→Definite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

If $x^{13} y^7 = (x + y)^{20},$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}$
→
Evaluate $:\begin{vmatrix}\text{x}+\lambda&\text{x}&\text{x}\\\text{x}&\text{x}+\lambda&\text{x}\\\text{x}&\text{x}&\text{x}+\lambda\end{vmatrix}$
→
If $\text{y}=\text{x}^3\log\text{x},$ Prove that $\frac{\text{d}^4\text{y}}{\text{dx}^4}=\frac{6}{\text{x}}$
→
Maximize $Z=3 x+2 y$ subject to constraints $x+2 y \leq 10,3 x+y \leq 15, x \geq 0, y \geq 0$ by using graphical method.
→
At what points on the following curves, is the tangent parallel to $x-$axis? $y = 12(x + 1)(x - 2)$ on $[-1, 2]$
→
Show that $\text{A}=\begin{bmatrix} 5 & 3 \\-1 & -2 \end{bmatrix}$ satisfies the equation $x^2 - 3x - 7 = 0.$ Thus, find $A^{-1}.$
→
Two factories decided to award their employees for three values of (a) adaptable to new techniques, (b) careful and alert in difficult situations and (c) keeping clam in tense situations, at the rate of ₹ x, ₹ y and ₹ z per person respectively. The first factory decided to honuor respectively 2, 4 and 3 employees with a total prize money of ₹ 29000. The second factory decided to honuor respectively 5, 2 and 3 employees with the prize money of ₹ 30500. If the three prizes per person together cost ₹ 9500, then
  1.  Represent the above situation by matrix equation and form linear equation using matrix multiplication.
  2. Solve this equation by matrix method.
  3. Which values are reflected in the questions?
→
$\text{if } \vec{\text{a}} = 2\hat{\text{i}} + \hat{\text{j}} - \hat{\text{k}}, \vec{\text{b}} = 4\hat{\text{i}} - 7\hat{\text{j}} + \hat{\text{k}}, \text{find a vector } \vec{\text{c}} \text{ such that } \vec{\text{a}} \times \vec{\text{c}} \text{ and } \vec{\text{a }} . \vec{\text{c}} = 6.$
→
Prove the following identities:
$\begin{vmatrix}\text{a}^3&2&\text{a}\\\text{b}^3&2&\text{b}\\\text{c}^3&2&\text{c}\end{vmatrix}$
$=2(\text{a}-\text{b})(\text{b}-\text{c})(\text{c}-\text{a})(\text{a}+\text{b}+\text{c})$
→
If $\text{xy}\log(\text{x}+\text{y})=1,$ prove that $\frac{\text{dx}}{\text{dx}}=-\frac{\text{y}(\text{x}^2\text{y}+\text{x}+\text{y})}{\text{x}(\text{xy}^2+\text{x}+\text{y})}$
→