Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[Let\ I = \int_0^1 \frac{1}{2 x^2 + x + 1} d\ x . Then, \]
\[I = \frac{1}{2} \int_0^1 \frac{1}{x^2 + \frac{x}{2} + \frac{1}{2}} d x\]
\[I = \frac{1}{2} \int_0^1 \frac{1}{\left( x^2 + \frac{x}{2} + \frac{1}{16} \right) - \frac{1}{16} + \frac{1}{2}} d\ x\]
\[ \Rightarrow I = \frac{1}{2} \int_0^1 \frac{1}{\left( x + \frac{1}{4} \right)^2 + \frac{7}{16}} dx\]
\[ \Rightarrow I = \frac{1}{2} \times \frac{4}{\sqrt{7}} \left[ \tan^{- 1} \left( \frac{x + \frac{1}{4}}{\frac{\sqrt{7}}{4}} \right) \right]_0^1 \]
\[ \Rightarrow I = \frac{2}{\sqrt{7}}\left( \tan^{- 1} \frac{5}{\sqrt{7}} - \tan^{- 1} \frac{1}{\sqrt{7}} \right)\]
APPEARS IN
संबंधित प्रश्न
Evaluate each of the following integral:
Prove that:
Evaluate each of the following integral:
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.
If \[\int\limits_0^a 3 x^2 dx = 8,\] write the value of a.
\[\int\limits_0^1 \left\{ x \right\} dx,\] where {x} denotes the fractional part of x.
The value of the integral \[\int\limits_{- 2}^2 \left| 1 - x^2 \right| dx\] is ________ .
The value of \[\int\limits_0^{\pi/2} \log\left( \frac{4 + 3 \sin x}{4 + 3 \cos x} \right) dx\] is
\[\int\limits_0^1 \left( \cos^{- 1} x \right)^2 dx\]
\[\int\limits_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} dx\]
\[\int\limits_0^{15} \left[ x^2 \right] dx\]
\[\int\limits_0^{\pi/2} \frac{\cos^2 x}{\sin x + \cos x} dx\]
\[\int\limits_0^\pi \frac{dx}{6 - \cos x}dx\]
\[\int\limits_0^4 x dx\]
\[\int\limits_{- 1}^1 e^{2x} dx\]
Using second fundamental theorem, evaluate the following:
`int_(-1)^1 (2x + 3)/(x^2 + 3x + 7) "d"x`
Evaluate the following:
`int_0^oo "e"^(- x/2) x^5 "d"x`
Choose the correct alternative:
`int_0^oo "e"^(-2x) "d"x` is
