Evaluate the following integrals: x^ -1 3 + x +2 [3] x dx - Vidyadip

Question

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

✓

Answer

$\int\Bigg(\int\frac{\text{x}^{-\frac{1}{3}}+\sqrt{\text{x}}+2}{\text{x}^{\frac{1}{3}}}\Bigg)\text{dx}$
$=\int\Bigg(\frac{\text{x}^{-\frac{1}{3}}}{\text{x}^{\frac{1}{3}}}+\frac{\text{x}^{\frac{1}{2}}}{\text{x}^{\frac{1}{3}}}+\frac{2}{\text{x}^{\frac{1}{3}}}\Bigg)\text{dx}$
$=\int\Big(\text{x}^{-\frac{2}{3}}+\text{x}^{\frac{1}{6}}+2\text{x}^{-\frac{1}{3}}\Big)\text{dx}$
$=\Bigg[\frac{\text{x}^{-\frac{2}{3}+1}}{-\frac{2}{3}+1}+\frac{\text{x}^{\frac{1}{6}+1}}{\frac{1}{6}+1}+2\frac{\text{x}^{-\frac{1}{3}+1}}{-\frac{1}{3}+1}\Bigg]$
$=\Bigg[\frac{\text{x}^{\frac{1}{3}}}{\frac{1}{3}}+\frac{\text{x}^{\frac{7}{6}}}{\frac{7}{6}}+3\text{x}^{\frac{2}{3}}\Bigg]+\text{C}$
$=3\text{x}^{\frac{1}{3}}+\frac{6}{7}\text{x}^{\frac{7}{6}}+3\text{x}^{\frac{2}{3}}+\text{C}$

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 Indefinite Integrals→Indefinite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Solve the differential equation $\bigg[\frac{\text{e}^{-2\sqrt{\text{x}}}}{\sqrt{\text{x}}}-\frac{\text{y}}{\sqrt{\text{x}}}\bigg]\frac{\text{dx}}{\text{dy}}=1(\text{x}\neq0).$

If the mean and variance of a binomial distribution are 4 and 3, respectively, find the probability of no success.

Let $R^+$ be the set of all non$-$negative real numbers. If $f : R^+ \rightarrow R^+$ and $g : R^+ \rightarrow R^+$ are defined as $f(x) = x^2$ and $\text{g(x)}=+\sqrt{\text{x}},$ find $fog$ and $gof.$ Are they equal functions?

If $\vec{\text{a}}$ and $\vec{\text{b}}$ are two vectors of the same magnitude inclined at an angle of 30°, such that $\vec{\text{a}}.\vec{\text{b}}=3,$ find $|\vec{\text{a}}|,\big|\vec{\text{b}}\big|.$

Evaluate the following integrals:
$\int\text{x}^2\text{e}^{\text{x}^3}\cos\big(\text{e}^{\text{x}^3}\big)\text{dx}$

Let $A = R - \{3\}$ and $B = R - \{1\}.$ Consider the function $f : A \rightarrow B$ defined by $f(x) = \left( {\frac{{x - 2}}{{x - 3}}} \right)$ Is f one-one and onto? Justify your answer.

Write the direction cosines of the line whose cartesian equations are 6x -2 = 3y + 1 =2z - 4.

Find $gof$ and $fog$ when $f : R \rightarrow R$ and $g : R \rightarrow R$ are defined by$: f(x) = 8x^3$ and $\text{g(x)}=\text{x}^\frac{1}{3}$

In a triangle OAC, if B is the mid-point of side AC and $\overrightarrow{\text{OA}} = \vec{\text{a}}, \overrightarrow{\text{OB}} = \vec{\text{b}},$ then was is $\overrightarrow{\text{OC}}$?

For a loaded die, the probabilities of outcome are given as under:
$P(1) = P(2) = 0.2, P(3) = P(5) = P(6) = 0.1$ and $P(4) = 0.3.$ the die is thrown two times. If the die were fair, determine whether or not the events $A$ and $B$ are independent.