Evaluate the following integrals: ^9_4 x (30-x^ 3 2 )^2 dx - Vidyadip

Question

Evaluate the following integrals:
$\int^\limits{9}_4\frac{\sqrt{\text{x}}}{\big(30-\text{x}^{\frac{3}{2}}\big)^2}\text{ dx}$

✓

Answer

Let $\text{I}=\int^\limits{9}_4\frac{\sqrt{\text{x}}}{\big(30-\text{x}^{\frac{3}{2}}\big)^2}\text{ dx}$ Then,
Let $\Big(30-\text{x}^{\frac{3}{2}}\Big)=\text{t}$ Then, $-\frac{3}{2}\sqrt{\text{x}}\text{ dx}=\text{dt}$
When $\text{x}=4,\text{t}=22$ and $\text{x}=9,\text{t}=3$
$\therefore\ \text{I}=\int^\limits{3}_{22}-\frac{2}{3}\frac{1}{\text{t}^2}\text{ dt}$
$\Rightarrow\text{I}=\frac{2}{3}\Big[\frac{1}{\text{t}}\Big]^3_{22}$
$\Rightarrow\text{I}=\frac{2}{3}\Big(\frac{1}{3}-\frac{1}{22}\Big)$
$\Rightarrow\text{I}=\frac{19}{99}$

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 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

Find the projection of $\overrightarrow{b} + \overrightarrow{c} $ on $\overrightarrow{a}$ where $\overrightarrow{a} = 2\hat{i} - 2\hat{j} + \hat{k}, \overrightarrow{b} = \hat{i} + 2\hat{j} - 2\hat{k} $ and $\overrightarrow{c} = 2\hat{i} - \hat{j} + 4\hat{k}.$

Prove that $y = \frac{{4\sin \theta }}{{\left( {2 + \cos \theta } \right)}} - \theta $ is an increasing function of $\theta$ in $\left[ {0,\frac{\pi }{2}} \right]$

Prove that the greatest integer function defined by $f\left( x \right) = \left[ x \right],0 < x < 3$ is not differentiable at x = 1 and x = 2.

Show that the Modulus Function f: R → R, given by f (x) = |x|, is neither one-one nor onto, where |x| is x, if x is positive or 0 and |x| is –x, if x is negative.

If $\text{y}=\log\Big(\sqrt{\text{x}}+\frac{1}{\sqrt{\text{x}}}\Big),$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\text{x}-1}{2\text{x}(\text{x}+1)}$

Evaluate the following integrals:
$\int\sin^5\text{x}\cos\text{x}\text{ dx}$

A die is thrown three times. Find $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)$ and $\text{P}\Big(\frac{\text{B}}{\text{A}}\Big)$, if
A = 4 appears on the third toss,
B = 6 and 5 appear respectively on first two tosses.

Examine the following functions for continuity.
$\text f(\text x)=\begin{vmatrix}\text x-5\end{vmatrix}$

If a function g = {(1, 1), (2, 3), (3, 5), (4, 7)} is described by $\text{g(x)}=\alpha\text{x}+\beta\alpha\text{x}+\beta,$ then find the values of $\alpha$ and $\beta.$

The mathematics department has 8 graduate assistants who are assigned to the same office. Each assistant is just as likely to study at home as in office. How many desks must there be in the office so that each assistant has a desk at least 90% of the time?