Evaluate the following integrals: ^1_0 (1 x -1 )dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^1_0\log\Big(\frac{1}{\text{x}}-1\Big)\text{dx}$

✓

Answer

Let $\text{I}=\int\limits^1_0\log\Big(\frac{1}{\text{x}}-1\Big)\text{dx}\ ....(\text{i})$
$=\int\limits^1_0\log\Big(\frac{1}{1-\text{x}}-1\Big)\text{dx}$ $\Big[\text{Using},\int\limits^{\text{a}}_0\text{f(x)}\text{dx}=\int\limits^{\text{a}}_0\text{f}(\text{a}-\text{x})\text{dx}\Big]$
$\text{I}=\int\limits^1_0\log\Big(\frac{\text{x}}{\text{x}-1}\Big)\text{dx}\ ....(\text{ii})$
Adding (i) and (ii)
$2\text{I}=\int\limits^1_0\log\Big(\frac{1-\text{x}}{\text{x}}\Big)\log\Big(\frac{1-\text{x}}{\text{x}}\Big)\text{dx}$
$=\int\limits^1_0\log\Big(\frac{1-\text{x}}{\text{x}}\times\frac{\text{x}}{1-\text{x}}\Big)\text{dx}$
$=\int\limits^1_0\log1\text{ dx}$
$=0$
Hence, $\text{I}=0$

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

An aeroplane can carry a maximum of 200 passengers. A profit of Rs.1000 is made on each executive class ticket and a profit of Rs.600 is made on each economy class ticket. The airline reserves atleast 20 seats for executive class. However, atleast 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximise the profit of the airline. What is the maximum profit?

Differentiate the following functions with respect to x:
$\tan^{-1}\Big\{\frac{\text{x}}{\sqrt{\text{a}^2-\text{x}^2}}\Big\},-\text{a}<\text{x}<\text{a}$

Find the absolute maximum and the absolute minimum value of the following functions in the given intervals:
$\text{f}(\text{x})=(\text{x}-2)\sqrt{\text{x}-1}\ \text{in}\ [1,9]$

Find the differential equation of all the parabolas with latus rectum '4a' and whose axes are parallel to x-axis.

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{c}^2(\text{x}^2+\text{y}^2)=\text{x}^2\text{y}^2\text{ at }\Big(\frac{\text{c}}{\cos\theta},\frac{\text{c}}{\cos\theta}\Big)$

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\sin^4\text{x}+\cos^4\text{x}\text{ on }\Big[0,\frac{\pi}{2}\Big]$

Find the area of the minor segment of the circle $x^2+ y^2 = a^2$ cut off by the line $\text{x}=\frac{\text{a}}{2}.$

Show that the function $f : R \rightarrow R$ defined by $f(x)=\frac{x}{x^2+1}, \forall x \in R$ is neither one$-$one nor onto.

Find the maximum and the minimum values, if any, without using derivaives of the following functions:f(x) = sin2x + 5 on R.

Use product $\begin{bmatrix}1&-1&2\\0&2&-3\\3&-2&4\end{bmatrix}\begin{bmatrix}-2&0&1\\9&2&-3\\6&1&-2\end{bmatrix}$ to solve the system of equations $x + 3z = 9, -x + 2y - 2z = 4, 2x - 3y + 4z = -3.$