_ ^ x^5 ;dx (1 + x^3 ) = - Vidyadip

MCQ

$\int_{}^{} {\frac{{{x^5}\;dx}}{{\sqrt {(1 + {x^3})} }} = } $

A
$\frac{2}{3}\sqrt {(1 + {x^3})} ({x^3} + 2)$
B
$\frac{2}{9}\sqrt {(1 + {x^3})} ({x^3} - 4)$
C
$\frac{2}{9}\sqrt {(1 + {x^3})} ({x^3} + 4)$
✓
$\frac{2}{9}\sqrt {(1 + {x^3})} ({x^3} - 2)$

✓

Answer

Correct option: D.

$\frac{2}{9}\sqrt {(1 + {x^3})} ({x^3} - 2)$

d
(d) Here ${x^5} = {x^3}{x^2}$ and differential coefficient of ${x^3}$ is $3{x^{2}}$ In order to remove fractional powers, we put
$1 + {x^3} = {t^2} \Rightarrow 3{x^2}dx = 2t.\,dt\,;$ Also ${x^3} = {t^2} - 1$
$I = \int_{}^{} {\frac{{({t^2} - 1)}}{t}\left( {\frac{2}{3}t\,dt} \right) = \frac{2}{3}\int_{}^{} {({t^2} - 1)\,dt} } $
$ = \frac{2}{3}\left( {\frac{{{t^3}}}{3} - t} \right) = \frac{2}{9}t\,({t^2} - 3)$= $\frac{2}{9}\sqrt {(1 + {x^3})} \,({x^3} - 2)$ $(\because \,{t^2} = 1 + {x^3})$

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 "M.C.Q (1 Marks)" from 7.1 inderfinite integral→7.1 inderfinite integral — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $ab + bc + ca = 0$ and $\left| {\,\begin{array}{*{20}{c}}{a - x}&c&b\\c&{b - x}&a\\b&a&{c - x}\end{array}\,} \right| = 0$, then one of the value of $x$ is

The value of $\int\limits^{\pi}_0\frac{1}{5+3\cos\text{x}}\text{ dx}$ is:

$\frac{\pi}{4}$
$\frac{\pi}{8}$
$\frac{\pi}{2}$
$0$

The domain of definition of the function $y(x)$ given by ${2^x} + {2^y} = 2$ is

Five persone entered the lift cabin on the ground floor of an 8 floor house. Suppose that each of them independently and with equal probability can leave the cabin at any flor beginning with the first, then the probability of all 5 persons leaving at different floors is,

Derivative of $e^{2 x}$ with respect to $e^x$, is

If $\vec{a}=2 \hat{i}+\hat{j}+2 \hat{k},$ then the value of $|\hat{ i } \times(\overrightarrow{ a } \times \hat{ i })|^{2}+|\hat{j} \times(\overrightarrow{ a } \times \hat{ j })|^{2}+|\hat{ k } \times(\overrightarrow{ a } \times \hat{ k })|^{2}$ is equal to

A value of $x$ for which $\sin \,\left( {{{\cot }^{ - 1}}\,\left( {1 + x} \right)} \right) = \cos \,\left( {{{\tan }^{ - 1}}\,x} \right)$, is

The value of $\left( {\vec a + 2\vec b - \vec c} \right).\left\{ {\left( {\vec a - \vec b} \right) \times \left( {\vec a - \vec b - \vec c} \right)} \right\}$ is equal to

The function $\text{f(x)}=|\cos\text{x}|$ is:

Everywhere continuous and differentiable.
Everywhere continuous but not differentiable at $(2\text{n}+1)\frac{\pi}{2},\text{n}\in\text{Z}$
Neither continuous nor differentiable at $(2\text{n}+1)\frac{\pi}{2},\text{n}\in\text{Z}$
None of these.

The corner points of the feasible region are A(0, 0), B(16, 0), C(8, 16) and D(0, 24). The minimum value of the objective function z = 300x + 190y is _______:

5440
4800
4560
0