Evaluate the following intregals: 1 x (2+ ) dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{1}{\text{x}\log\text{x}(2+\log\text{x})}\text{ dx}$

✓

Answer

Let $\int\frac{1}{\text{x}\log\text{x}(2+\log\text{x})}=\frac{\text{A}}{\text{x}\log\text{x}}+\frac{\text{B}}{\text{x}(2+\log\text{x})}$
$\Rightarrow1=\text{A}(2+\log\text{x})+\text{B}\log\text{x}$
Put x = 1
$\Rightarrow1=2\text{A}\Rightarrow\text{A}=\frac{1}{2}$
Put x = 10^-2
$\Rightarrow1=-2\text{B}\Rightarrow\text{B}=-\frac{1}{2}$
Thus,
$\text{I}=\frac{1}{2}\int\frac{\text{dx}}{\text{x}\log\text{x}}+\Big(-\frac{1}{2}\Big)\int\frac{\text{dx}}{\text{x}(2+\log\text{x})}$
$=\frac{1}{2}\log|\log\text{x}|-\frac{1}{2}\log|2+\log\text{x}|+\text{C}$
$\text{I}=\frac{1}{2}\log\Big|\frac{\log\text{x}}{2+\log\text{x}}\Big|+\text{C}$

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

Similar questions

Solve the following system of equations by matrix method:
2x + 6y = 2
3x - z = -8
2x - y + z = -3

If f(x) = x³ + ax² + bx + c has a maximum at x = -1 and minimum at x = 3. Determine a, b and c.

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

In order to supplement daily diet, a person wishes to take X and Y tablets. The contents (in milligrams per tablet) of iron, calcium and vitamins in X and Y are given as below:

Tablets	Iron	Calcium	Vitamin
X	6	3	2
Y	2	3	4

The person needs to supplement at least 18 milligrams of iron, 21 milligrams of calcium and 16 milligrams of vitamins. The price of each tablet of X and Y is ₹ 2 and ₹1 respectively. How many tablets of each type should the person take in order to satisfy the above requirement at the minimum cost? Make an LPP and solve graphically.

Differentiate $\cos^{-1}(4\text{x}^3-3\text{x})$ with respect to $\tan^{-1}\Big(\frac{\sqrt{1-\text{x}^2}}{\text{x}}\Big),$ if $\frac{1}{2}<\text{x}<1$

Evaluate the following integrals:
$\int_{0}^\limits{{\pi}}\frac{1}{5+3\cos\text{x}}\text{ dx}$

A company manufactures two types of toys A and B. Type A requires 5 minutes each for cutting and 10 minutes each for assembling. Type B requires 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours available for cutting and 4 hours available for assembling in a day. The profit is Rs. 50 each on type A and Rs. 60 each on type B. How many toys of each type should the company manufacture in a day to maximize the profit?

Using integration, find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x² + y² = 32.

Find $x$, if $[x-5-1]\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3\end{array}\right]\left[\begin{array}{l}x \\ 4 \\ 1\end{array}\right]=O$.

Evaluate the following integral:
$\int\frac{1}{\sqrt{(2-\text{x})^2-1}}\text{ dx}$