Evaluate the following integrals: ^ _0 | |dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^\pi_0\cos\text{x}|\cos\text{x}|\text{dx}$

✓

Answer

Consider $\text{f(x)}=\cos\text{x}|\cos\text{x}|$
Now,
$\text{f}(\pi-\text{x})=\cos(\pi-\text{x})|\cos(\pi-\text{x})|$
$=-\cos\text{x}|-\cos\text{x}|=-\cos\text{x}|\cos\text{x}|$
$=-\text{f(x)}$
$\therefore\ \int\limits^\pi_0\cos\text{x}|\cos\text{x}|\text{dx}=0$ $\begin{bmatrix}\int\limits^{2\text{a}}_0\text{f(x)}\text{dx}=\begin{cases}\int\limits^{2\text{a}}_0\text{f(x)}\text{dx},&\text{ if }\text{ f}(2\text{a}-\text{x})=\text{f(x)}\\0,&\text{ if }\text{ f}(2\text{a}-\text{x})=\text{f(x)}\end{cases}\end{bmatrix}$

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 Question" from INTEGRALS→INTEGRALS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Tickets are numbered from 1 to 10. Two tickets are drawn one after the other at random. Find the probability that the number on one of the tickets is a multiple of 5 and on the other a multiple of 4.

Find the distance of the point P(-1, -5, -10) from the point of intersection of the line joining the points A(2, -1, 2) and B(5, 3, 4) with the plane x - y + z = 5.

If the vertices A, B, C of a triangle ABC are the points with position vectors $\text{a}_1\hat{\text{i}}+\text{a}_2\hat{\text{j}}+\text{a}_3\hat{\text{k}},\ \text{b}_1\hat{\text{i}}+\text{b}_2\hat{\text{j}}+\text{b}_3\hat{\text{k}},\ \text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}}$ respectively, what are the vectors determined by its sides? Find the length of these vectors.

Find a vector in the direction of vector $5\hat{i}-\hat{j}+2\hat{k}$ which has magnitude 8 units.

Find the projection of $\vec{b}+\vec{c}$ on $\vec{a}$ where $\vec{a}=\hat{i}+2\hat{j}+\hat{k},\text{ }\vec{b}=\hat{i}+3\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}+\hat{k}.$

The items produced by a company contain 10% defective items. Show that the probability of getting 2 defective items in a sample of 8 items is $\frac{28\times9^6}{10^8}$.

Prove that $\int_0^{\pi / 2} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x=\frac{\pi}{4}$

The volume of a cube is increasing at the rate of $9\ cm^3/ \sec.$ How fast is the surface area increasing when the length of an edge is $10\ cm?$

Write the following in the simplest form:
$\sin\Big\{2\tan^{-1}\sqrt{\frac{1-\text{x}}{1+\text{x}}}\Big\}$

Find the equation of a passing through the point (-1, -1, 2) and perpendicular to the planes 3x + 2y - 3z = 1 and 5x - 4y + z = 5.