Evaluate the following definite integrals: _ 0 ^ 2 ^3x dx - Vidyadip

Question

Evaluate the following definite integrals:
$\int\limits_{0}^{\frac{\pi}{2}}\cos^3\text{x}\text{ dx}$

✓

Answer

Let $\text{I}=\int_{0}^\limits{\frac{\pi}{2}}\cos^3\text{x}\text{ dx}$ Then,
$\text{I}=\int_{0}^\limits{\frac{\pi}{2}}\cos^3\text{x}\cos\text{x}\text{ dx}$
$\Rightarrow\text{I}=\int_{0}^\limits{\frac{\pi}{2}}\big(1-\sin^2\text{x}\big)\cos\text{x}\text{ dx}$
Let $\text{u}=\sin\text{x},\text{ du}=\cos\text{x dx}$
$\Rightarrow\text{I}=\int(1-\text{u}^2)\text{du}$
$\Rightarrow\text{I}=\Big[\text{u}-\frac{\text{u}^3}{3}\Big]$
$\Rightarrow\text{I}=\Big[\sin\text{x}-\frac{\sin^3\text{x}}{3}\Big]^{\frac{\pi}{2}}_0$
$\Rightarrow\text{I}=1-\frac{1}{3}-0$
$\Rightarrow\text{I}=\frac{2}{3}$

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 lines $\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7}$ and $\frac{x-2}{1}=\frac{y+4}{3}=\frac{z-6}{5}$ intersect each other. Find also the coordinates of the point of intersection.

Solve the following differential equation:
$(\text{x}-\text{y})\frac{\text{dy}}{\text{dx}}=\text{x + 2y}$

Find the particular solution of the differential equation
$\tan x.\frac{\text{dy}}{\text{dx}}=2x \tan x+x^2-\text{y};(\tan x\neq0)\text{given that y}=0 \ \text{when x}=\frac{\pi}{2}$

Prove that:
$\begin{vmatrix}\text{a}^2&2\text{ab}&\text{b}^2\\\text{b}^2&\text{a}^2&2\text{ab}\\2\text{ab}&\text{b}^2&\text{a}^2\end{vmatrix}=(\text{a}^3+\text{b}^3)^2$

Find the distance of the point (1, -2, 4) from plane passing throuhg the point (1, 2, 2) and perpendicular of the planes x - y + 2z = 3 and 2x - 2y + z + 12 = 0

Verify Rolle's theorem for the following function on the indicated intervals $f(x) = x(x^{ }- 4)^2$ on the interval $[0, 4]$

Solve the following systems of homogeneous linear equations by matrix method:

$x + y - 6z = 0$

$x - y + 2z = 0$

$-3x + y + 2z = 0$

Compute the adjoint of the following matrices:$\begin{bmatrix}2 & 0 & -1 \\5 & 1 & 0 \\ 1 & 1 & 3 \end{bmatrix}$
Verify that $\ce{(adj A)A = |A| I = A (adj A)}$ for the above matrices.

Find the cartesian form of the equations of the following planes.
$\vec{\text{r}}=(\hat{\text{i}}-\hat{\text{j}})+\text{s}(-\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}})+\text{t}(\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}})$

If $\text{y}=\log\frac{\text{x}^2+\text{x}+1}{\text{x}^2-\text{x}+1}+\frac{2}{\sqrt{3}}\tan^{-1}\Big(\frac{\sqrt{3}\text{x}}{1-\text{x}^2}\Big),$ find $\frac{\text{dy}}{\text{dx}}$