By using the properties of definite integrals, evaluate the integ… - Vidyadip

Question

By using the properties of definite integrals, evaluate the integral $\int\limits_0^{\frac{\pi }{2}} {{{\cos }^2}xdx} $

✓

Answer

Let $I = \int\limits_0^{\frac{\pi }{2}} {{{\cos }^2}xdx} $…(i)
$= \int\limits_0^{\frac{\pi }{2}} {{{\cos }^2}\left( {\frac{\pi }{2} - x} \right)dx} $
$\left[ {\because \int\limits_0^a {f\left( x \right)dx = \int\limits_0^a {f\left( {a - x} \right)dx = } } } \right]$
$ \Rightarrow I= \int\limits_0^{\frac{\pi }{2}} {{{\sin }^2}xdx} $ …(ii)
Adding equation (i) and (ii),
$2I = \int\limits_0^{\frac{\pi }{2}} {\left( {{{\cos }^2}x + {{\sin }^2}x} \right)dx} $
$= \int\limits_0^{\frac{\pi }{2}} {1dx} $
$\Rightarrow 2I = \left( x \right)_0^{\frac{\pi }{2}}$
$ \Rightarrow 2I = \frac{\pi }{2}$
$ \Rightarrow I = \frac{\pi }{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 "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

Find the shortest distance between the lines $l_1$ and $l_2$ whose vector equations are
$\vec r = \hat i + \hat j + \lambda (2\hat i - \hat j + \hat k) ...(1) $
and $\vec r = 2\hat i + \hat j - \hat k + \mu (3\hat i - 5\hat j + 2\hat k) ...(2)$

Probality of solving specific problem independently by $A$ and $B$ are $\frac{1}{2}$ and $\frac{1}{3}$ respectively. If both try to solve the problem independently. Find the probability that the problem is solved.

Evaluate the following integrals:
$\int^\limits6_{-6}\big|\text{x}+2\big|\text{dx}$

Check the commutativity and associativity of the following binary operations: $'\odot\ '$ on $Q$ defined by $a \odot b = a^2 + b^2$ for all $a, b \in Q.$

Examine the differentialiblilty of the function f defined by $\text{f(x)}=\begin{cases}2\text{x}+3 & \text{if}-3\leq\text{x}\leq-2\\\text{x}+1 & \text{if} -2\leq\text{x}\leq0\\\text{x}+2&\text{if}\ 0\leq\text{x}\leq1\end{cases}$

Find the angle between the plane:
2x - y + z = 4 and x + y + 2z = 3

If $\text{A}=\text{diag}\begin{pmatrix}2&-5&9\end{pmatrix},\text{ B}=\text{diag}\begin{pmatrix}1&1&-4\end{pmatrix}$ and $\text{C}=\text{diag}\begin{pmatrix}-6&3&4\end{pmatrix},$ find.
$2\text{A}+3\text{B}-5\text{C}$

If $f : R \rightarrow (0, 2)$ defined by $\text{f(x)}=\frac{\text{e}^{\text{x}}-\text{e}^{-\text{x}}}{\text{e}^{\text{x}}+\text{e}^{-\text{x}}}+1$ is invertible, find $f^{-1}$.

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$\text{f}(\text{x})=\sqrt{\text{x}^2-4}\text{ on }[2,4]$

If $x^y=y^x$, then find $\frac{d y}{d x}$.