Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[\text{We have}, \]
\[I = \int_0^1 \frac{2x}{1 + x^2} d x\]
\[\text{Putting} 1 + x^2 = t\]
\[ \Rightarrow 2x\ dx = dt\]
\[\text{When } x \to 0; t \to 1\]
\[\text{And } x \to 1; t \to 2\]
\[ \therefore I = \int_1^2 \frac{d t}{t}\]
\[ = \left[ \log_e \left| t \right| \right]_1^2 \]
\[ = \log_e 2 - \log_e 1\]
\[ = \log_e 2 - 0\]
\[ = \log_e 2\]
APPEARS IN
संबंधित प्रश्न
\[\int\limits_1^4 f\left( x \right) dx, where f\left( x \right) = \begin{cases}7x + 3 & , & \text{if }1 \leq x \leq 3 \\ 8x & , & \text{if }3 \leq x \leq 4\end{cases}\]
Evaluate the following integral:
If f is an integrable function, show that
\[\int\limits_{- a}^a f\left( x^2 \right) dx = 2 \int\limits_0^a f\left( x^2 \right) dx\]
Prove that:
Evaluate each of the following integral:
If \[\int\limits_0^1 \left( 3 x^2 + 2x + k \right) dx = 0,\] find the value of k.
\[\int\limits_0^{\pi/4} \sin 2x \sin 3x dx\]
\[\int\limits_0^{\pi/2} \frac{1}{1 + \tan^3 x} dx\]
\[\int\limits_0^\pi \frac{x}{a^2 \cos^2 x + b^2 \sin^2 x} dx\]
\[\int\limits_0^{15} \left[ x^2 \right] dx\]
\[\int\limits_0^{\pi/2} \frac{x}{\sin^2 x + \cos^2 x} dx\]
\[\int\limits_0^2 \left( 2 x^2 + 3 \right) dx\]
Using second fundamental theorem, evaluate the following:
`int_(-1)^1 (2x + 3)/(x^2 + 3x + 7) "d"x`
Evaluate the following using properties of definite integral:
`int_0^1 log (1/x - 1) "d"x`
Choose the correct alternative:
`int_0^oo x^4"e"^-x "d"x` is
Evaluate `int (x^2"d"x)/(x^4 + x^2 - 2)`
Evaluate `int (x^2 + x)/(x^4 - 9) "d"x`
The value of `int_2^3 x/(x^2 + 1)`dx is ______.
