Evaluate the following integrals: _ 0 ^ 2 x x+2 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits_{0}^{2}\text{x}\sqrt{\text{x}+2}\text{ dx}$

✓

Answer

$\int_{0}^\limits{2}\text{x}\sqrt{\text{x}+2}\text{ dx}$
Let $\text{x}+2=\text{t}^2\Rightarrow\text{dx}=2\text{tdt}$
When $\text{x}=0,\text{t}=\sqrt{2}$ and when $\text{x}=2,\text{t}=2$
$\therefore\ \int_{0}^\limits{2}\text{x}\sqrt{\text{x}+2}\text{ dx}=\int_{\sqrt{2}}^\limits{2}\big(\text{t}^2-2\big)\sqrt{\text{t}^2}2\text{t dt}$
$=2\int_{\sqrt{2}}^\limits{2}\big(\text{t}^2-2\big)\text{t}^2\text{ dt}$
$=2\int_{\sqrt{2}}^\limits{2}\big(\text{t}^4-2\text{t}^2\big)\text{dt}$
$=2\Big[\frac{\text{t}^5}{5}-\frac{2\text{t}^3}{3}\Big]^2_\sqrt{2}$
$=2\Big[\frac{32}{5}-\frac{16}{3}-\frac{4\sqrt{2}}{5}+\frac{4\sqrt{2}}{3}\Big]$
$=2\Big[\frac{96-80-12\sqrt{2}+20\sqrt{2}}{15}\Big]$
$=2\Big[\frac{16+8\sqrt{2}}{15}\Big]$
$=\frac{16\big(2+\sqrt{2}\big)}{15}$
$=\frac{16\sqrt{2}\big(\sqrt{2}+1\big)}{15}$

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 "4 Marks" from Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

For the matrix $\text{A}=\begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3 \end{bmatrix}.$ Show that A^-3 - 6A² + 5A + 11I₃ = 0 Hence, find A^-1.

A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/ cutting machine and a sprayer. It takes 2 hours on grinding/ cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/ cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/ cutting machine for at the most 12 hours. The profit from the sale of a lamp is Rs 5 and that from a shade is Rs 3. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximise his profit?

Find the values of a and b so that the function $\text{f(x)}\begin{cases}\text{x}^2+3\text{x}+\text{a}, & \text{if x}\leq1\\\text{bx}+2, & \text{if x} > 1\end{cases}$ is differentiable at each $\text{x}\in\text{R}.$

Let f: W → W be defined as f(n) = n – 1, if n is odd and f (n) = n + 1, if n is even. Show that f is invertible. Find the inverse of f. Here, W is the set of all whole numbers.

Find the distance between the point (-1, -5, -10) and the point of intersection of the line $\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 2}{12}$ and the plane $x - y + z = 5.$

Evaluate the following intregals:
$\int\frac{\cos\text{x}}{(1-\sin\text{x})^3(2+\sin\text{x})}\ \text{dx}$

Find the probability that the sum of the numbers showing on two dice is 8, given that at least one die does not show five.

The two adjecent sides of a parallelogram are $2\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}$ and $\hat{\text{i}}-2\hat{\text{j}}-3\hat{\text{k}}.$ Find the unit vector parallel to one of its diagonals. Also, find its area.

Find the equation of the curve which passes through the point (1, 2) and the distance between the foot of the ordinate of the point of contact and the point of intersection of the tangent with x-axis twice abscissa of the pont of contact.

For each of the differential equations in find a particular solution satisfying the given condition:
$(\text{x}^3+\text{x}^2+\text{x}+1) \frac{\text{dy}}{\text{dx}} = 2\text{x}^2+\text{x; y} =1 \ \text{when x = 0}$