Evaluate: _2^4 (x^2+x ) 2 x+1 d x - Vidyadip

Question

Evaluate: $\int_2^4 \frac{\left(x^2+x\right)}{\sqrt{2 x+1}} d x$

✓

Answer

$(d) :$ We have, $\int_2^4 \frac{\left(x^2+x\right)}{\sqrt{2 x+1}} d x$
Integrating by parts, we get
$\int_2^4 \frac{\left(x^2+x\right)}{\sqrt{2 x+1}} d x=\left[\left(x^2+x\right) \cdot \sqrt{2 x+1}\right]_2^4-\int_2^4(2 x+1) \cdot \sqrt{2 x+1} d x$
$=(60-6 \sqrt{5})-\int_2^4(2 x+1)^{3 / 2} d x$
$=(60-6 \sqrt{5})-\frac{1}{5} \cdot\left[(2 x+1)^{5 / 2}\right]_2^4$
$=(60-6 \sqrt{5})-\left(\frac{243}{5}-5 \sqrt{5}\right)=\left(\frac{57}{5}-\sqrt{5}\right)=\left(\frac{57-5 \sqrt{5}}{5}\right)$

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

Similar questions

The derivative of the function $\cot^{-1}\Big|(\cos2\text{x})^{\frac{1}{2}}\Big|\text{ at }\text{x}=\frac{\pi}{6}$ is:

$\Big(\frac{2}{3}\Big)^\frac{1}{2}$
$\Big(\frac{1}{3}\Big)^\frac{1}{2}$
$3^\frac{1}{2}$
$6^\frac{1}{2}$

The number of solutions of the system of inequations $x+2 y \leq 3,3 x+4 y \geq 12, x \geq 0, y \geq 1$ is

A coin is tossed 10 times. The probability of getting exactly six heads is:

$\frac{512}{513}$
$\frac{105}{512}$
$\frac{100}{153}$
$\text{ }^{10}\text{C}_6$

Let $\text{f(x)=}\begin{cases}\frac{\text{x}-4}{|\text{x}-4|}+\text{a},&\text{if }\text{ x} < 4\\\text{a}+\text{b},&\text{if }\text{ x} =4\\\frac{\text{x}-4}{|\text{x}-4|}+\text{b},&\text{if }\text{ x} > 4\end{cases}$ Then, f(x) is continus at x = 4 when:

a = 0, b = 0
a = 1, b = 1
a = -1, b = 1
a = 1, b = -1

The pressure P and volume V of a gas are connected by the relation $\text{PV}^{\frac{1}{4}}=$ constant. The percentage increase in the pressure corresponding to a deminition of $\frac{1}{2}\%$ in the volume is:

$\frac{1}{2}\%$
$\frac{1}{4}\%$
$\frac{1}{8}\%$
$\text{None of these}$

If A is a singular matrix, then adj A is:

Non-singular.
Singular.
Symmetric.
Not defined.

10 persons are seated around a round table. What is the probability that 4 particular persons are always seated together?

$\int\text{x}\sec\text{x}^2\text{ dx}$ is equal to:

$\frac{1}{2}\log\big(\sec\text{x}^2+\tan\text{x}^2\big)+\text{C}$
$\frac{\text{x}^2}{2}\log\big(\sec\text{x}^2+\tan\text{x}^2\big)+\text{C}$
$2\log\big(\sec\text{x}^2+\tan\text{x}^2\big)+\text{C}$
none of these.

The probability that an automobile will be stolen and found within one week is 0.0006. The probability that an automobile will be stolen is 0.0015. The probability that a stolen automobile will be found in one week is:

0.3
0.4
0.5
0.6

Choose the correct answer in Exercises:If $\frac{\text{d}}{\text{dx}}\text{f}\text{(x)}=4\text{x}^3-\frac{3}{\text{x}^4}$such that $\text{f}(2)=0.$Then $\text{f}\text{(x)}$ is

$\text{x}^4+\frac{1}{\text{x}^3}-\frac{129}{8}$
$\text{x}^3+\frac{1}{\text{x}^4}+\frac{129}{8}$
$\text{x}^4+\frac{1}{\text{x}^3}+\frac{129}{8}$
$\text{x}^3+\frac{1}{\text{x}^4}-\frac{129}{8}$