Question
Evaluate the following integrals:
$\int\frac{\log\big(1+\frac{1}{\text{x}}\big)}{\text{x}(1+\text{x})}\text{dx}$
$\int\frac{\log\big(1+\frac{1}{\text{x}}\big)}{\text{x}(1+\text{x})}\text{dx}$
✓
Answer
Let I $=\int\frac{\log\big(1+\frac{1}{\text{x}}\big)}{\text{x}(1+\text{x})}\text{dx}\ .....(1)$
Let $\log\Big(1+\frac{1}{\text{x}}\Big)=\text{t}$ then,
$\text{d}\Big[\log\Big(1+\frac{1}{\text{x}}\Big)\Big]=\text{dt}$
$\Rightarrow\frac{1}{1+\frac{1}{\text{x}}}\times\frac{-1}{\text{x}^2}\text{dx}=\text{dt}$
$\Rightarrow\frac{1}{\frac{\text{x}+1}{\text{x}}}\times\frac{-1}{\text{x}^2}\text{dx}=\text{dt}$
$\Rightarrow\frac{-\text{x}}{\text{x}^2(\text{x}+1)}\text{dx}=-\text{dt}$
$\Rightarrow\frac{\text{dx}}{\text{x}(\text{x}+1)}=-\text{dt}$
Putting $\log\Big(1+\frac{1}{\text{x}}\Big)=\text{t}$ and $\frac{\text{dx}}{\text{x}(\text{x}+1)}=-\text{dt}$ in equation (1), we get
$\text{I}=\int\text{tx}-\text{dt}$
$=-\frac{\text{t}^2}{2}+\text{C}$
$=-\frac{1}{2}\Big[\log\Big(1+\frac{1}{\text{x}}\Big)\Big]^2+\text{C}$
$\therefore\text{I}=-\frac{1}{2}\Big[\log\Big(1+\frac{1}{\text{x}}\Big)\Big]^2+\text{C}$
Let $\log\Big(1+\frac{1}{\text{x}}\Big)=\text{t}$ then,
$\text{d}\Big[\log\Big(1+\frac{1}{\text{x}}\Big)\Big]=\text{dt}$
$\Rightarrow\frac{1}{1+\frac{1}{\text{x}}}\times\frac{-1}{\text{x}^2}\text{dx}=\text{dt}$
$\Rightarrow\frac{1}{\frac{\text{x}+1}{\text{x}}}\times\frac{-1}{\text{x}^2}\text{dx}=\text{dt}$
$\Rightarrow\frac{-\text{x}}{\text{x}^2(\text{x}+1)}\text{dx}=-\text{dt}$
$\Rightarrow\frac{\text{dx}}{\text{x}(\text{x}+1)}=-\text{dt}$
Putting $\log\Big(1+\frac{1}{\text{x}}\Big)=\text{t}$ and $\frac{\text{dx}}{\text{x}(\text{x}+1)}=-\text{dt}$ in equation (1), we get
$\text{I}=\int\text{tx}-\text{dt}$
$=-\frac{\text{t}^2}{2}+\text{C}$
$=-\frac{1}{2}\Big[\log\Big(1+\frac{1}{\text{x}}\Big)\Big]^2+\text{C}$
$\therefore\text{I}=-\frac{1}{2}\Big[\log\Big(1+\frac{1}{\text{x}}\Big)\Big]^2+\text{C}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Integrate the following w.r.t. x:
→$\frac{1}{2 \cos x+3 \sin x}$
A company produces mixers and food processors. Profit on selling one mixer and one food processor is ₹ 2,000/- and ₹ 3,000/- respectively. Both the products are processed through three Machines A, B, C. The time required in hours by each product and total time available in hours per week on each machine are as follows :
How many mixers and food processors should be produced to maximize the profit?
Evaluate the following integrals:$\int^\limits{\frac{\pi}{2}}_0\cos^5\text{x dx}$
→Show that the cone of the greatest volume which can be inscribed in a given spher has an altitude equal to 2/3 of the diameter of the sphere.
Evaluate the following definite integrals:$\int_{0}^\limits{\frac{\pi}{2}}\cos^3\text{x}\text{ dx}$
→Find the equation of the containing the line $\frac{\text{x}+1}{-3}=\frac{\text{y}-3}{2}=\frac{\text{z}+2}{1}$ and the point $(0, 7, -7)$ and show that the line $\frac{\text{x}}{1}=\frac{\text{y}-7}{-3}=\frac{\text{z}+7}{2}$ also lies in the same plane.
→Evaluate the following integrals:$\int\limits^1_{-1}\log\Big(\frac{2-\text{x}}{2+\text{x}}\Big)\text{dx}$
→Find the area of the region between the circles $x^2 + y^2 = 4$ and $(x - 2)^2 + y^2 = 4.$
→$\int(2\text{x}^2+3)\sqrt{\text{x+2}}\text{dx}$
→Find the intervals in which the following functions are increasing or decreasing.
$f(x) = 8 + 36x + 3x^2 -2x^3$
→$f(x) = 8 + 36x + 3x^2 -2x^3$