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

Question

Evaluate the following integrals:
$\int\sqrt{\text{x}}\Big(\text{x}^3-\frac{2}{\text{x}}\Big)\text{dx}$

✓

Answer

$\int\sqrt{\text{x}}\Big(\text{x}^3-\frac{2}{\text{x}}\Big)\text{dx}$
$=\int\Big(\text{x}^{\frac{7}{2}}-\frac{2}{\sqrt{\text{x}}}\Big)\text{dx}$
$=\int\Big(\text{x}^{\frac{7}{2}}-{2}{\text{x}^{-\frac{1}{2}}}\Big)\text{dx}$
$=\frac{\text{x}^{\frac{7}{2}+1}}{\frac{7}{2}+1}-2\frac{\text{x}^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+\text{C}$
$=\frac{2}{9}\text{x}^{\frac{9}{2}}-4\text{x}^{\frac{1}{2}}+\text{C}$
$=\frac{2}{9}\text{x}^{\frac{9}{2}}-4\sqrt{\text{x}}+\text{C}$

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

If the vertices A, B and C of $\triangle\text{ABC}$ have position vectors (1, 2, 3), (-1, 0, 0) and (0, 1, 2), respectively, what is the magnitude of $\angle\text{ABC}?$

Let $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}},\vec{\text{d}}$ be the position vectors of the four distinct points A, B, C, D. If $\vec{\text{b}}-\vec{\text{a}}=\vec{\text{c}}-\vec{\text{d}}$, then show that ABCD is a parallelogram.

Evalute the following integrals: $\int\frac{10\text{x}^9+10^\text{x}\log_\text{e}10}{10^\text{x}+\text{x}^{10}}\text{dx}$

Find the maximum value of $\sin \theta+\cos \theta$.

If $\vec{\text{a}}$ and $\vec{\text{b}}$ are perpendicular vectors, $\big|\vec{\text{a}}+\vec{\text{b}}\big|=3$ and $|\vec{\text{a}}|=5,$ find the value of $\big|\vec{\text{b}}\big|.$

In the following, determine the values of constants involved in the definition so that the given function is continuous:
$\text{f(x)}=\begin{cases}\frac{\sin2\text{x}}{5\text{x}},&\text{if }\text{ x}\neq0\\3\text{k},&\text{if }\text{ x}=0\end{cases}$

Prove that
$\text{L.H.S}=\sin\Big\{\tan^{-1}\frac{1-\text{x}^2}{2\text{x}}+\cos^{-1}\frac{1-\text{x}^2}{1+\text{x}^2}\Big\}=1$

Evaluate the following integrals:$\int2\text{x}^3\text{e}^{\text{x}^{2}}\text{dx}$

$\int\frac{2\text{x}+3}{(\text{x}-1)^2}\text{dx}$

Evaluate the following integrals:
$\int^\limits4_1\text{f(x)}\text{dx},$ Where $\text{f(x)}=\begin{cases}4\text{x}+3,&\text{if }\ 1\leq\text{x}\leq2\\3\text{x}+5,&\text{if }\ 2\leq\text{x}\leq4\end{cases}$