Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[I = \int_0^1 \frac{2x}{5 x^2 + 1} d x + \int_0^1 \frac{3}{5 x^2 + 1} d x\]
\[ \Rightarrow I = \frac{1}{5} \int_0^1 \frac{10x}{5 x^2 + 1} d x + 3 \int_0^1 \frac{1}{\left( \sqrt{5}x \right)^2 + 1^2} d x\]
\[ \Rightarrow I = \frac{1}{5} \left[ \log \left( 5 x^2 + 1 \right) \right]_0^1 + \frac{3}{\sqrt{5}} \left[ \tan^{- 1} \left( \sqrt{5}x \right) \right]_0^1 \]
\[ \Rightarrow I = \frac{1}{5} \log 6 + \frac{3}{\sqrt{5}} \tan^{- 1} \sqrt{5}\]
APPEARS IN
संबंधित प्रश्न
Evaluate the following integral:
If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that \[\int_a^b xf\left( x \right)dx = \frac{a + b}{2} \int_a^b f\left( x \right)dx\]
If f is an integrable function, show that
Evaluate each of the following integral:
Evaluate each of the following integral:
Solve each of the following integral:
\[\int\limits_0^1 \left\{ x \right\} dx,\] where {x} denotes the fractional part of x.
The value of \[\int\limits_0^\pi \frac{x \tan x}{\sec x + \cos x} dx\] is __________ .
The value of \[\int\limits_0^{\pi/2} \cos x\ e^{\sin x}\ dx\] is
\[\int\limits_0^1 \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) dx\]
\[\int\limits_0^1 \frac{1 - x}{1 + x} dx\]
\[\int\limits_0^\pi \sin^3 x\left( 1 + 2 \cos x \right) \left( 1 + \cos x \right)^2 dx\]
\[\int\limits_{\pi/6}^{\pi/2} \frac{\ cosec x \cot x}{1 + {cosec}^2 x} dx\]
\[\int\limits_0^3 \left( x^2 + 1 \right) dx\]
Choose the correct alternative:
`int_(-1)^1 x^3 "e"^(x^4) "d"x` is
Choose the correct alternative:
Using the factorial representation of the gamma function, which of the following is the solution for the gamma function Γ(n) when n = 8 is
Choose the correct alternative:
Γ(n) is
Evaluate `int sqrt((1 + x)/(1 - x)) "d"x`, x ≠1
Evaluate `int (x^2 + x)/(x^4 - 9) "d"x`
What is the meaning of an indefinite integral?
What is an antiderivative of \[x\]?
Evaluate \[\int_{0}^{1}x\,dx\].
If velocity changes with time, what does a definite integral give over a time interval?
