Evaluate the following integrals: ^ 2 _ - 2 x ^2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^\frac{\pi}{2}_{-\frac{\pi}{2}}\text{x}\cos^2\text{x}\text{ dx}$

✓

Answer

We have,
$\text{I}=\int\limits^\frac{\pi}{2}_{-\frac{\pi}{2}}\text{x}\cos^2\text{x}\text{ dx}$
Let $\text{f}(\text{x})=\text{x}\cos^2\text{x}$
$\Rightarrow \text{f}(-\text{x})= (-\text{x})\cos^2(-\text{x})$
$= - \text{x}\cos^2\text{x}$
$\therefore \text{f}(-\text{x})=-\text{f}(\text{x})$
i.e., f(x) is odd function.
We know that $\int\limits^\text{a}_{-\text{a}}\text{f}(\text{x})\text{dx} = 0, $ if f(x) is odd function.
$\therefore\ \text{I}=\int\limits^\frac{\pi}{2}_{-\frac{\pi}{2}}\text{x}\cos^2\text{x}\text{ dx}=0$

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 "3 Marks" from Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\text{A}\begin{bmatrix}1&0&1\\0&1&2\\0&0&4\end{bmatrix},$ then show that |3A| = 27|A|.

Discuss the continuity of the function f(x) at the point $\text{x}=\frac{1}{2}$ where
$\text{f}\text{(x)}=\begin{cases}\text{x}, & 0\leq\text{x} < \frac{1}{2}\\\frac{1}{2},&\text{x}=\frac{1}{2}\\1-\text{x}, &\frac{1}{2}< \text{x}\leq 1\end{cases}$

Write the direction cosines of the line whose cartesian equations are 6x -2 = 3y + 1 =2z - 4.

The line $\vec{\text{r}}=\hat{\text{i}}+\lambda(2\hat{\text{i}}-\text{m}\hat{\text{j}}-3\hat{\text{k}})$ is parallel to the plane $\vec{\text{r}}\cdot(\text{m}\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}})=4.$ Find m.

A can hit a target 3 times in 6 shots, B : 2 times in 6 shots and C : 4 times in 4 shots. They fix a volley. What is the probability that at least 2 shots hit?

Write the equation of the normal to the curve $\text{y}=\text{x}+\sin\text{x}\cos\text{x}\text{ at }\text{x}=\frac{\pi}{2}.$

The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (Marginal revenue). If the total revenue (in rupees) recieved from the sale of x units of a product is given by R(x) = 3x² + 36x + 5, find the marginal revenue, when x = 5, and write which value does the question indicate.

Let A and B be sets. Show that f: A × B → B × A such that f (a, b) = (b, a) is bijective function.

Differentiate the following functions with respect to x:
$\log\Big\{\cot\Big(\frac{\pi}{4}+\frac{\pi}{2}\Big)\Big\}$

For what value of k is the function
$\text{f}\text{(x)}=\begin{cases}\frac{\sin2\text{x}}{\text{x}}, & \text{x} \neq 0\\\text{k}, &\text{x} = 0\end{cases}$ continuous at x = 0.