Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS5 Marks
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

Find the perpendicular distence of the point (1, 0, 0) from the line $\frac{\text{x}-1}{2}=\frac{\text{y}+1}{-3}=\frac{\text{z}+10}{8}.$ Also, find the coordinates of the perpendicular and the equation of the perpendicular.
→
Let S be the set of all real numbers except -1 and let '*' be an operation defined by a * b = a + b + ab for all a, b ∈ S. Determine whether '*' is a binary operation on S. If yes, check its commutativity and associativity. Also, solve the equation (2 * x) * 3 = 7.
→
Evaluate the following integrals:
$\int\tan^{-1}(\sqrt{\text{x}})\text{dx}$
→
Evaluate the following integrals:
$\int\limits_{0}^{\frac{\pi}{4}}\big(\text{a}^2\cos^2\text{x}+\text{b}^2\sin^2\text{x}\big)\text{dx}$
→
The probability that A hits a target is $\frac{1}{3}$ and the probability that B hits it, is $\frac{2}{5}$, What is the probability that the target will be hit, if each one of A and B shoots at the target?
→
If $x^y + y^x = (x + y)^{x+y}$, find $\frac{\text{dy}}{\text{dx}}$
→
If $\text{x}=\cos\text{t}(3-2\cos^2\text{t}),\text{y}\sin\text{t}(3-2\sin^2\text{t})$ find the value of $\frac{\text{dy}}{\text{dx}}\text{ at t}=\frac{\pi}{4}$
→
Find the area of the bounded by $\text{y}=\sqrt{\text{x}}$ and $y^2 = x.$
→
Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,
$\text{f}(\text{x})=\frac{2}{\text{x}}-\frac{2}{\text{x}^{2}}, \text{x}>0$
→
If $\text{y}=\text{e}^{\text{x}}\cos\text{x},$ Prvoe that $\frac{\text{dy}}{\text{dx}}=\sqrt{2}\text{e}^\text{x}.\cos\Big(\text{x}+\frac{\pi}{4}\Big)$
→