Advertisements
Advertisements
प्रश्न
Find `int_0^1 x(tan^-1x) "d"x`
उत्तर
I = `int_0^1x(tan^-1x)^2 "d"x`
Integrating by parts, we have
I = `x^2/2[(tan^-1x)^2]_0^1 - 1/2 int_0^1 x^2 * 2 (tan^-1x)/(1 + x^2) "d"x`
= `pi^2/32 - int_0^1 x^2/(1 + x) * tan^-1 x"d"x`
= `pi^2/32 - 1_1`, where I1 = `int_0^1 x^2/(1 + x^2) tan^-1 x"d"x`
Now I1 = `int_0^1 (x^2 + 1 - 1)/(1 + x^2) tan^-1x "d"x`
= `int_0^1 tan^-1 x"d"x - int_0^1 1/(1 + x^2) tan^-1 x"d"x`
= `"I"_2 - 1/2 ((tan^-1x)^2)_0^1`
= `"I"_2 - pi^2/32`
Here I2 = `int_0^1 tan^-1 x"d"x = (x tan^-1x)_0^1 - int_0^1 x/(1 + x^2) "d"x`
= `pi/4 - 1/2(log|1 + x^2|)_0^1`
= `pi/4 - 1/2 log2`
Thus I2 = `pi/4 - 1/2 log 2 - pi^2/32`
Therefore, I = `pi^2/32 - pi/4 + 1/2 log2 + pi^2/32`
= `pi^2/16 - pi/4 + 1/2 log2`
= `(pi^2 - 4pi)/16 + log sqrt(2)`.
APPEARS IN
संबंधित प्रश्न
Prove that:
`int sqrt(x^2 - a^2)dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c`
Integrate the function in x sin 3x.
Integrate the function in x log 2x.
Integrate the function in x sin-1 x.
Evaluate the following : `int sin θ.log (cos θ).dθ`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `sqrt(5x^2 + 3)`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Choose the correct options from the given alternatives :
`int (1)/(cosx - cos^2x)*dx` =
Choose the correct options from the given alternatives :
`int [sin (log x) + cos (log x)]*dx` =
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`
`int"e"^(4x - 3) "d"x` = ______ + c
`int 1/sqrt(x^2 - 8x - 20) "d"x`
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
Solve: `int sqrt(4x^2 + 5)dx`
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Evaluate:
`int e^(logcosx)dx`
If u and v are two differentiable functions of x, then prove that `intu*v*dx = u*intv dx - int(d/dx u)(intv dx)dx`. Hence evaluate: `intx cos x dx`
Evaluate the following:
`intx^3e^(x^2)dx`
Evaluate:
`inte^x "cosec" x(1 - cot x)dx`
