Evaluate the following integrals: ^ _0x ^2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{\pi}_0\text{x}\cos^2\text{x dx}$

✓

Answer

Let $\text{I}=\int\limits^{\pi}_0\text{x}\cos^2\text{x dx}\ ...(\text{i})$
$=\int\limits^{\pi}_0(\pi-\text{x})\cos^2(\pi-\text{x})\text{dx}$
$=\int\limits^{\pi}_0(\pi-\text{x})\cos^2\text{x}\text{ dx}\ ...(\text{ii})$
Adding (i) and (ii) we get
$2\text{I}=\int\limits^{\pi}_0(\text{x}+\pi-\text{x})\cos^2\text{x}\text{ dx}$
$=\int\limits^{\pi}_0\pi\cos^2\text{x}\text{ dx}$
$=\pi\int\limits^{\pi}_0\frac{1+\cos2\text{x}}{2}\text{ dx}$
$=\frac{\pi}{2}\int\limits^{\pi}_0\big(1+\cos2\text{x}\big)\text{dx}$
$=\frac{\pi}{2}\Big[\text{x}+\frac{\sin2\text{x}}{2}\Big]^{\pi}_0$
$=\frac{\pi}{2}(\pi-0)$
Hence, $\text{I}=\frac{\pi^2}{4}$

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

Show that the vectors $2 \hat{i}-\hat{j}+\hat{k}$, $\hat{i}-3 \hat{j}-5 \hat{k}$ and $3 \hat{i}-4 \hat{j}-4 \hat{k}$ from the vertices of a right angled triangle.

Differentiate $\frac{\text{x}}{\sin\text{x}}$ w.r.t. $\sin\text{x}.$

Differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{y}=0,\text{y}(0)=2,\text{y}'(0)=0$ Function $\text{y}=\text{e}^\text{x}+\text{e}^{-\text{x}}$

Using differentials, find the approximate values of the following:
$\sqrt{401}$

Evaluate: $\int\frac{\text{x}^{2}}{(\text{x}^{2} + 4)(\text{x}^{2} + 9 )}\text{dx}.$

Using integration find the area of the region bounded by the curves $\text{y} = \sqrt{4 - \text{x}^{2}}, \text{x}^{2} + \text{y}^{2} \text{4x} = 0$ and the x-axis.

Maximise and Minimise Z = 3x – 4y subject to:
$\text{x}-2\text{y}\leq0$
$-3\text{x}+\text{y}\leq4$
$\text{x}-\text{y}\leq6$
$\text{x},\text{y}\geq0$

A company makes 3 model of calculators: A, B and C at factory I and factory II. The company has orders for at least 6400 calculators of model A, 4000 calculator of model B and 4800 calculator of model C. At factory I, 50 calculators of model A, 50 of model B and 30 of model C are made every day; at factory II, 40 calculators of model A, 20 of model B and 40 of model C are made everyday. It costs Rs. 12000 and Rs. 15000 each day to operate factory I and II, respectively. Find the number of days each factory should operate to minimise the operating costs and still meet the demand.

If $\text{y}=\text{cosec}^{-1}\text{x},\text{x}>1$ prove that $\text{x}(\text{x}^2-1)\frac{\text{d}^2\text{y}}{\text{dx}^2}+(2\text{x}^2-1)\frac{\text{dy}}{\text{dx}}=0.$

Evaluate the following integrals:
$\int\limits^{\pi}_0\text{x}\cos^2\text{x dx}$