Evaluate the following integrals: _ 1 ^ 2 1 x (1+ )^2 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits_{1}^{2}\frac{1}{\text{x}\big(1+\log\text{x}\big)^2}\text{ dx}$

✓

Answer

Let $1+\log\text{x}=\text{t}$
Differentiating w.r.t. x, we get
$\frac{1}{\text{x}}\text{ dx}=\text{dt}$
Now, $\text{x}=1\Rightarrow\text{t}=1$
$=\text{x}=2\Rightarrow\text{t}=1+\log2$
$\therefore\ \int_{1}^\limits{2}\frac{1}{\text{x}\big(1+\log\text{x}\big)^2}\text{ dx}=\int^\limits{1+\log2}_1\frac{\text{dt}}{\text{t}^2}$
$=\Big[\frac{-1}{\text{t}}\Big]^{1+\log2}_1$
$=\bigg[\frac{-1}{1+\log2}+1\bigg]$
$=\bigg[\frac{-1+1+\log2}{1+\log2}\bigg]$
$=\bigg[\frac{\log2}{1+\log2}\bigg]$ $\big[\because\log\text{e}=1\big]$
$=\frac{\log2}{\log\text{e}+\log2}$ $\big[\log\text{a}+\log\text{b}=\log\text{ab}\big]$
$=\frac{\log2}{\log2\text{e}}$

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 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

The equation of tangent at (2, 3) on the curve $y^{2} = \text{ax}^{3} + \text{b is y = 4x - 5}. $ Find the value of a and b.

If $\text{f}\text{(x)}=\begin{cases}\frac{\text{x}-4}{\text{|x}-4|}+\text{a}, &\text{if x} <4\\\text{a}+\text{b},&\text{if x}=4\\\frac{\text{x}-4}{\text{|x}-4|}+\text{b}, & \text{if x}>4\end{cases}$ is continuous at $x = 4$. Find $a, b$.

Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1%.

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

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,
$\text{f}(\text{x})=\frac{\text{x}}{2}+\frac{2}{\text{x}}, \text{x}>0$

Find the derivative of the function $f(x)$ given by $f(x) = (1 + x) (1 + x^2) (1 + x^4) (1 + x^8)$ and hence find $f(1).$

If $\vec{\text{a}}\text{ and }\vec{\text{b}}$ are two non-collinear vectors having the same initial point. What are the vectors represented by $\vec{\text{a}}+\vec{\text{b}}\text{ and }\vec{\text{a}}-\vec{\text{b}}$.

Evaluate the following integrals:
$\int\frac{\text{x}^2}{1-\text{x}^4}\ \text{dx}$

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{x}=\text{at}^2,\text{y}=2\text{at at}\text{ t}=1$

If $\text{f}\text{(x)}=\begin{cases}\frac{\text{x}^2-1}{\text{x}-1},& \text{for }\text{ x}\neq1 \\2,&\text{for }\text{ x}=1\end{cases}$ Find whether f (x) is continuous at x = 1.