Advertisements
Advertisements
प्रश्न
Evaluate the definite integral:
`int_0^2 (6x +3)/(x^2 + 4)` dx
Advertisements
उत्तर
Let `I = int_0^2 (6x + 3)/(x^2 + 4) dx`
`= int_0^2 (6x) /(x^2 + 4) dx + int_0^2 3/ (x^2 + 4) dx`
`= 3 int_0^2 (2x)/(x^2 + 4) dx + [3 xx 1/2 tan^-1 x/2]_0^2`
Let `I_1 = 3 int_0^2 (2x)/ (x^2 + 4) dx`
Put x2 + 4 = t
⇒ 2x dx = dt
When x = 0
⇒ t = 4
When x = 2
⇒ t = 8
`= 3 int_4^8 dt/t = [3 log t]_4^8`
= 3 (log 8 - log 4)
= 3 log 2
⇒ `I = 3 log 2 + 3/2 [tan^-1 1 tan^-1 0]`
`= 3 log 2 + 3/2 xx pi/4 = 3 log 2 + (3pi)/8`
APPEARS IN
संबंधित प्रश्न
Evaluate : `∫_0^(π/2)(sin^2 x)/(sinx+cosx)dx`
Evaluate :`int_0^(pi/2)(2^(sinx))/(2^(sinx)+2^(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_0^(pi/2) cos 2x dx`
Evaluate the definite integral:
`int_4^5 e^x dx`
Evaluate the definite integral:
`int_0^1 dx/sqrt(1-x^2)`
Evaluate the definite integral:
`int_2^3 dx/(x^2 - 1)`
Evaluate the definite integral:
`int_0^(pi/2) cos^2 xdx`
Evaluate the definite integral:
`int_2^3 (xdx)/(x^2 + 1)`
Evaluate the definite integral:
`int_0^1 (2x + 3)/(5x^2 + 1) dx`
Evaluate the definite integral:
`int_0^1 x e^(x^2) dx`
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`
`int_1^(sqrt3)dx/(1+x^2) ` equals:
`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
Evaluate:
`int_0^π(sin^4x + cos^4x)dx`
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?
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 expression correctly applies the limits to a definite integral?
Which function is an antiderivative of \[\sin^3 2t\cos 2t?\]
Evaluate \[\int_0^{\frac\pi4}\sin^3 2t\cos 2t\,dt.\]
