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 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

Evaluate the following integrals:
$\int\limits^2_1\frac{1}{\text{x}(1+\log\text{x})^2}\text{ dx}$

Differentiate the following function with respect to x:$(\log\text{x})^{x} + \text{x}^{\log\text{x}}.$

Solve the following differential equation: $(\text{x}^{2} - \text{y}^{2}) \text{dx} + \text{2xy dy =0}$ given that $y = 1$ when $x = 1$

Verify Rolle's theorem for the following function on the indicated intervals$\text{f}(\text{x})=\cos2\Big(\text{x}-\frac{\pi}{4}\Big)\text{ on }\Big[0,\frac{\pi}{2}\Big]$

If $\text{A}=\begin{bmatrix} 3\\5\\2\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&0&4\end{bmatrix},$ verify that $(AB)^T = B^TA^T$.

Differentiate the following functions with respect to x:
$10^{\log\sin\text{x}}$

Solve the differential equation: $(\text{x}+1)\frac{\text{dy}}{\text{dx}}=2\text{e}^{-\text{y}}-1;\ \text{y(0)}=0.$

Verify mean value theorem for the function:
$\text{f(x)}=\text{x}^3-2\text{x}^2-\text{x}+3\text{ in }[0,1].$

A curve is such that the length of the perpendicular from the origin on the tangent at any point P of the curve is equal to the abscissa of P. Prove that the differential equation of the curve is $\text{y}^{2}-2\text{xy}\frac{\text{dy}}{\text{dx}}-\text{x}^{2}=0$ and hence find the curve.

The prices of three commodities $P, Q$ and $R$ are Rs. $x, y$ and $z$ per unit respectively. A purchases $4$ units of $R$ and sells $3$ units of $P$ and $5$ units of $Q$. $B$ purchases $3$ units of $Q$ and sells $2$ units of $P$ and $1$ unit of $R$. $C$ purchases $1$ unit of $P$ and sells $4$ units of $Q$ and $6$ units of $R$. In the process $A, B$ and $C$ earn $Rs. 6000 , Rs. 5000$ and Rs. $13000$ respectively. If selling the units is positive earning and buying the units is negative earnings, find the price per unit of three commodities by using matrix method.