Advertisements
Advertisements
Question
Evaluate the following : `int_0^1 (1/(1 + x^2))sin^-1((2x)/(1 + x^2))*dx`
Solution
Let I = `int_0^1 (1/(1 + x^2))sin^-1((2x)/(1 + x^2))*dx`
Put x = tan t, i.e. t = tan–1x
∴ dx = sec2t dt
When x = 1, t = tan–11 = `pi/(4)`
When x = 0, t = tan–1 0 = 0
∴ I = `int_0^(pi/4) (1/(1 + tan^2t))sin^-1 ((2tan t)/(1 + tan^2t))sec^2t*dt`
= `int_0^(pi/4) (1)/(sec^2t) sin^-1 (sin 2t) sec^2t*dt`
= `int_0^(pi/4) 2t*dt`
= `2int_0^(pi/4)t*dt`
= `2[(t^2)/2]_0^(pi/4)`
= `2[pi/(32) - 0]`
= `pi^2/(16)`.
APPEARS IN
RELATED QUESTIONS
Evaluate : `int_2^3 (1)/(x^2 + 5x + 6)*dx`
Evaluate:
`int_0^(pi/4) sqrt(1 + sin 2x)*dx`
Evaluate : `int_(-1)^1 (1)/(a^2e^x + b^2e^(-x))*dx`
Evaluate : `int_1^3 (cos(logx))/x*dx`
Evaluate the following:
`int_((-pi)/2)^(pi/2) log((2 + sin x)/(2 - sin x)) * dx`
Evaluate the following : `int_0^1 (log(x + 1))/(x^2 + 1)*dx`
Evaluate the following : `int_0^1 t^2 sqrt(1 - t)*dt`
Evaluate the following : `int_0^(pi/2) [2 log (sinx) - log (sin 2x)]*dx`
Evaluate the following : `int_(-2)^(3) |x - 2|*dx`
Evaluate the following : if `int_a^a sqrt(x)*dx = 2a int_0^(pi/2) sin^3x*dx`, find the value of `int_a^(a + 1)x*dx`
Evaluate the following definite integrals: If `int_0^"a" (2x + 1)*dx` = 2, find the real value of a.
Evaluate the following definite integral:
`int_1^2 (3x)/((9x^2 - 1))*dx`
Evaluate the following integrals : `int_(-9)^9 x^3/(4 - x^2).dx`
Evaluate the following integrals : `int_0^"a" x^2("a" - x)^(3/2)*dx`
Choose the correct alternative :
`int_2^3 x/(x^2 - 1)*dx` =
Choose the correct alternative :
If `int_0^"a" 3x^2*dx` = 8, then a = ?
Choose the correct alternative :
`int_2^3 x^4*dx` =
Choose the correct alternative :
`int_0^2 e^x*dx` =
Fill in the blank : `int_0^1 dx/(2x + 5)` = _______
State whether the following is True or False : `int_"a"^"b" f(x)*dx = int_(-"b")^(-"a") f(x)*dx`
State whether the following is True or False : `int_"a"^"b" f(x)*dx = int_"a"^"b" f("t")*dt`
Solve the following:
`int_0^1 e^(x^2)*x^3dx`
Solve the following : `int_1^2 e^(2x) (1/x - 1/(2x^2))*dx`
Choose the correct alternative:
`int_(-2)^3 1/(x + 5) "d"x` =
`int_1^2 x^2 "d"x` = ______
`int_0^"a" 4x^3 "d"x` = 81, then a = ______
Evaluate `int_1^"e" 1/(x(1 + log x)^2) "d"x`
Evaluate `int_1^2 (3x)/((9x^2 - 1)) "d"x`
`int_0^1 tan^-1 ((2x - 1)/(1 + x - x^2))` dx = ?
Evaluate the following definite intergrals.
`int_1^3 logx* dx`
Evaluate the following definite intergral:
`int_-2^3 1/(x+5) dx`
Solve the following.
`int_1^3x^2 logx dx`
`int_0^(π/2) (sin^2 x.dx)/(1 + cosx)^2` = ______.
Evaluate the following definite intergral:
`int_4^9 1/sqrt(x)dx`
If `int_((-pi)/4) ^(pi/4) x^3 * sin^4 x dx` = k then k = ______.
Evaluate the following definite intergral:
`int_1^2(3x)/((9x^2-1))dx`
Solve the following.
`int_0^1 e^(x^2) x^3 dx`
Evaluate the following definite integral:
`int_-2^3 1/(x+5).dx`
Evaluate the following definite intergral:
`int_1^3 log x dx`
Solve the following.
`int_1^3x^2 logx dx`
