Advertisements
Advertisements
प्रश्न
Evaluate the definite integral:
`int_0^1 (2x + 3)/(5x^2 + 1) dx`
Advertisements
उत्तर
Let `I = int_0^1 (2x + 3)/(5x^2 + 1) dx`
`= int_0^1 (2x)/(5x^2 + 1) dx + int_0^1 3/(5x^2 + 1) dx`
`= 1/5 int_0^1 (10 x)/(5x^2 + 1) dx + 3/5 int_0^1 1/(x^2 + 1/sqrt5) dx`
`= 1/5 [log (5x^2 + 1)]_0^1 + 3/5 . 1/(1 sqrt5) [tan^-1 (x/ 1)/sqrt5]_0^1`
`= 1/5 [log 6 - log 1] + 3/sqrt5 [tan^-1 sqrt5 - tan^-1 0]`
`= 1/5 log 6 + 3/sqrt5 tan^-1 sqrt5`
APPEARS IN
संबंधित प्रश्न
If `int_0^h1/(2+8x^2)dx=pi/16 `then find the value of h.
Evaluate : `∫_0^(π/2)(sin^2 x)/(sinx+cosx)dx`
Evaluate the definite integral:
`int_(-1)^1 (x + 1)dx`
Evaluate the definite integral:
`int_2^3 1/x dx`
Evaluate the definite integral:
`int_1^2 (4x^3 - 5x^2 + 6x + 9) dx`
Evaluate the definite integral:
`int_0^(pi/2) cos 2x dx`
Evaluate the definite integral:
`int_0^(pi/4) tan x dx`
Evaluate the definite integral:
`int_(pi/6)^(pi/4) cosec x dx`
Evaluate the definite integral:
`int_2^3 (xdx)/(x^2 + 1)`
Evaluate the definite integral:
`int_1^2 (5x^2)/(x^2 + 4x + 3)`
Evaluate the definite integral:
`int_0^(pi/4) (2 sec^2 x + x^3 + 2) dx`
Evaluate the definite integral:
`int_0^pi (sin^2 x/2 - cos^2 x/2) dx`
Evaluate the definite integral:
`int_0^2 (6x +3)/(x^2 + 4)` dx
Evaluate the definite integral:
`int_0^1 (xe^x + sin (pix)/4)`
`int_0^(2/3) dx/(4+9x^2)` equals:
Evaluate : \[\int\frac{x \cos^{- 1} x}{\sqrt{1 - x^2}}dx\] .
`int_1^sqrt(3) (dx)/(1 + x^2)` equals
`int_6^(2/3) (dx)/(4 + 9x^2)` equals
Hence evaluate:
`int_(-2π)^(2π) (sin^4x + cos^4x)/(1 + e^x)dx`
What does the Fundamental Theorem of Integral Calculus connect?
How can a definite integral be evaluated using the Fundamental Theorem of Integral Calculus?
If \(f\) is continuous on an interval, which expression defines the area function?
What does \[A(x)=\int_a^x f(t)\,dt\] give?
If \(f\) is continuous on \([a,b]\) and \[A(x)=\int_a^x f(t)\,dt,\] what is \(A'(x)\) for every \(x\) in \((a,b)\)?
Which condition permits direct use of the First Fundamental Theorem and the Second Fundamental Theorem on \([a,b]\)?
Why is there no need to write the constant of integration \(C\) while evaluating a definite integral?
Which expression correctly applies the limits to a definite integral?
For \[\int_0^{\frac\pi4}\sin^3 2t\cos 2t\,dt,\] which substitution and differential relation are correct?
Which function is an antiderivative of \[\sin^3 2t\cos 2t?\]
