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

Write the angle between the lines 2x = 3y = -z and 6x = -y = -4z.

Define a binary operation$ *$ on the set $\{0, 1, 2, 3, 4, 5\}$ as $\text{a}*\text{b}=\begin{cases}\text{a + b},&\text{if a + b}<6\\\text{a + b}-6&\text{if a + b}\geq6\end{cases}$
Show that zero is the identity for this operation and each element a of the set is invertible with $6 – a$ being the inverse of a.

Show that $\text{f}(\text{x})=\text{e}^{\frac{1}{\text{x}}},\text{x}\neq0$ is a decreasing function for all $\text{x}\neq0.$

Let * be a binary operation on $Q_0 ($set of non$-$zero rational numbers) defined by $\text{a}\ ^* \ \text{b}=\frac{\text{ab}}{5}$ for all $\text{a, b}\in\text{Q}_0.$ Show that * is commutative as well as associative. Also, find its identity element if it exists.

Differentiate the following functions with respect to x:
$\log(\cos\text{x}^2)$

A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale otherwise it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.

If `$\text{y}=(\cos\text{x})^{(\cos\text{x})^{(\cos\text{x})\dots\infty}},$ show that $\frac{\text{dy}}{\text{dx}}=\frac{\text{y}^2\tan\text{x}}{\text{y}\log\cos\text{x}-1}.$

Evaluate:
$\cos\Big(\tan^{-1}\frac{3}{4}\Big)$

For each of the differential equation given in find the general solution: $\text{x}\log\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=\frac{2}{\text{x}}\log\text{x}$

Find the vector equation of the line passing through the point (2, -1, -1) which is parallel to the line 6x - 2 = 3y +1 =2z - 2.