_ 0 ^ 1 x (1+2x)dx - Vidyadip

Question

$\int\limits_{0}^{1}\text{x}\log(1+2\text{x})\text{dx}$

✓

Answer

Let $\text{I}=\int\limits_{0}^{1}\text{x}\log(1+2\text{x})\text{dx}$
$=\Bigg[\log(1+2\text{x})\frac{\text{x}^2}{2}\Bigg]_{0}^{1}-\int\frac{1}{1+2\text{x}}\cdot2\cdot\frac{\text{x}^2}{2}\text{dx}$
$=\frac{1}{2}\big[\text{x}^2\log(1+2\text{x})\big]_{0}^{1}-\int\frac{\text{x}^2}{1+2\text{x}}\text{dx}$
$=\frac{1}{2}[\log3-0]-\Bigg[\int\limits_{0}^{1}\bigg(\frac{\text{x}}{2}-\frac{\frac{\text{x}}{2}}{1+2\text{x}}\bigg)\text{dx}\Bigg]$
$=\frac{1}{2}\log3-\frac{1}{2}\int\limits^{1}_{0}\text{x dx}+\frac{1}{2}\int\limits^{1}_{0}\frac{\text{x}}{1+2\text{x}}\text{dx}$
$=\frac{1}{2}\log3-\frac{1}{2}\bigg[\frac{\text{x}^2}{2}\bigg]^{1}_{0}+\frac{1}{2}\int\limits^{1}_{0}\frac{\frac{1}{2}(2\text{x}+1-1)}{(2\text{x}+1)}\text{dx}$
$=\frac{1}{2}\log3-\frac{1}{2}\Big[\frac{1}{2}-0\Big]+\frac{1}{4}\int^1_0\text{dx}-\frac{1}{4}\int^1_0\frac{1}{1+2\text{x}}\text{dx}$
$=\frac{1}{2}\log3-\frac{1}{4}+\frac{1}{4}\big[\text{x}\big]^{1}_{0}-\frac{1}{8}\big[\log|(1+2\text{x})|\big]^{1}_{0}$
$=\frac{1}{2}\log3-\frac{1}{4}+\frac{1}{4}-\frac{1}{8}\big[\log3-\log1\big]$
$=\frac{1}{2}\log3-\frac{1}{8}\log3$
$=\frac{3}{8}\log3$

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 "5 Marks Questions" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Using properties of determinants show that $\begin{vmatrix} 1 & 1 & \text{1 + x} \\ 1 & \text{1 + y} & 1 \\ \text{1 + z} & 1 & 1 \end{vmatrix} = \text{xyz + yz + zx + xy}.$

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\frac{\text{dx}}{1+\tan\text{x}}$

Give that vectors $\overrightarrow{\text{a}}, \overrightarrow{\text{b}}, \overrightarrow{\text{c}}.$ form a triangle such that $\overrightarrow{\text{a}} = \overrightarrow{\text{b}} + \overrightarrow{\text{c}}.$ Find p, q, r, s such that area of triangle is $5\sqrt{6}$ where $\overrightarrow{\text{a}} = \text{p }\hat{i} + \text{q }\hat{j} + \text{r }\hat{k} = \overrightarrow{\text{b}} = \text{s }\hat{i} + \text{3 }\hat{j} + \text{4 }\hat{k} \text{ and} \overrightarrow{\text{c}} = \text{3 }\hat{i} + \hat{j} - \text{2}\hat{k}.$

A box of constant volume c is to be twice as long as it is wide. The material on the top and four sides cost three times as much per square metre as that in the bottom. What are the most economic dimensions?

Two cards are drawn simultaneosly from a well shuffled deck of 52 cards. Find the probability distribution of the number of the successes, when getting a spade is considered a success.

40% students of a college reside in hostel and the remaining reside outside. At the end of the year, 50% of the hostelers got A grade while from outside students, only 30% got A grade ain the examination. At the ed of the year, a student of the college was chosen at random and was found to have gotten A grade. What is the probability that the selected student was a hosteler?

Differentiate the following functions with respect to x:
$\frac{\text{e}^{2\text{x}}+\text{e}^{-2\text{x}}}{\text{e}^{2\text{x}}-\text{e}^{-2\text{x}}}$

Find the equation of the plane that is perpendicular to the plane $5x + 3y + 6z + 8 = 0$ and which contains the line of intersection of the planes $x + 2y + 3z - 4 = 0, 2x + y - z + 5 = 0$.

Find the area of the ragion bounded by $x^2= 4ay$ and its latusrectum.

Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=(\tan\text{x})^{\cot\text{x}}+(\cot\text{x})^{\tan\text{x}}$