Evaluate the definite integral _0^ ( ^2 x 2 - ^2 x 2 )dx - Vidyadip

Question

Evaluate the definite integral $\int\limits_0^\pi {\left( {{{\sin }^2}\frac{x}{2} - {{\cos }^2}\frac{x}{2}} \right)dx} $

✓

Answer

$\int\limits_0^\pi {\left( {{{\sin }^2}\frac{x}{2} - {{\cos }^2}\frac{x}{2}} \right)dx} $ $= \int\limits_0^\pi {\left( {\frac{{1 - \cos x}}{2} - \frac{{1 + \cos x}}{2}} \right)dx} $
$= \int\limits_0^\pi {\left( {\frac{{1 - \cos x - 1 - \cos x}}{2}} \right)dx} $
$ = \int\limits_0^\pi {\left( {\frac{{ - 2\cos x}}{2}} \right)dx} $
$ = - \int\limits_0^\pi {\cos xdx} $
$ = - \left( {\sin x} \right)_0^\pi $
$= - \left( {\sin \pi - \sin {0^o}} \right)$
= -(0 - 0) = 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 "2 Marks" from Integrals→Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Given a non empty set X, consider P (X) which is the set of all subsets of X.
Define the relation R in P (X) as follows:
For subsets A, B in P (X), ARB if and only if A $\subset$ B. Is R an equivalence relation on P (X)? Justify your answer.

Let * be a binary operation defined by a * b = 3a + 4b − 2. Find 4 * 5.

Write the projection of the vector $7\hat{\text{i}}+\hat{\text{j}}-4\hat{\text{k}}$ on the vector $2\hat{\text{i}}+6\hat{\text{j}}+3\hat{\text{k}}.$

Evaluate the following integrals:
$\int\frac{1}{3\sqrt{\text{x}^2}}\text{dx}$

In a competition A, B and C are participating. The probability that A wins is twice that of B, the probability that B wins is twice that of C. Find the probability that A losses.

Find the absolute maximum value and the absolute minimum value of the function:
$f(x)=(x-1)^{2}+3, x \in[-3,1]$

If A is a square matrix such that A² = A, then write the value of 7A − (I + A)³, where I is the identity matrix.

If $\vec{\text{a}}$ and $\vec{\text{b}}$ are unit vectors, then write the value of $\big|\vec{\text{a}}\times\vec{\text{b}}\big|^2+\big(\vec{\text{a}}.\vec{\text{b}}\big)^2.$

If C_ij is the cofactor of the element a_ij of the matrix $\text{A}=\begin{bmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{bmatrix},$ then write the value of a₃₂C₃₂.

If $\vec{\text{a}}$ is a unit vector such that $\vec{\text{a}}\times\hat{\text{i}}=\hat{\text{j}},$ find $\vec{\text{a}}.\hat{\text{i}}.$