Advertisements
Advertisements
प्रश्न
Evaluate the following : `int_0^1 (1/(1 + x^2))sin^-1((2x)/(1 + x^2))*dx`
उत्तर
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
संबंधित प्रश्न
Evaluate:
`int_0^(pi/4) sqrt(1 + sin 2x)*dx`
Evaluate:
`int_0^(pi/2) sqrt(cos x) sin^3x * dx`
Evaluate : `int_0^(pi/4) sec^4x*dx`
Evaluate : `int_1^3 (cos(logx))/x*dx`
Choose the correct option from the given alternatives :
If `[1/logx - 1/(logx)^2]*dx = a + b/(log2)`, then
Choose the correct option from the given alternatives :
`int_1^2 (1)/x^2 e^(1/x)*dx` =
Evaluate the following : `int_0^(pi/2) cosx/(3cosx + sinx)*dx`
Evaluate the following : `int_0^1 1/(1 + sqrt(x))*dx`
Evaluate the following : `int_0^1 t^5 sqrt(1 - t^2)*dt`
Evaluate the following : `int_0^pi (sin^-1x + cos^-1x)^3 sin^3x*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 integrals : `int_1^2 sqrt(x)/(sqrt(3 - x) + sqrt(x))*dx`
Choose the correct alternative :
`int_"a"^"b" f(x)*dx` =
State whether the following is True or False : `int_0^"a" f(x)*dx = int_"a"^0 f("a" - x)*dx`
Solve the following : `int_2^3 x/((x + 2)(x + 3))*dx`
Solve the following : `int_2^3 x/(x^2 + 1)*dx`
Solve the following : `int_(-4)^(-1) (1)/x*dx`
Solve the following : `int_0^1 (1)/(sqrt(1 + x) + sqrt(x))dx`
Solve the following : `int_1^2 (5x^2)/(x^2 + 4x + 3)*dx`
State whether the following statement is True or False:
`int_2^3 x/(x^2 + 1) "d"x = 1/2 log 2`
State whether the following statement is True or False:
`int_0^(2"a") "f"(x) "d"x = int_0^"a" "f"(x) "d"x + int_0^"a" "f"("a" - x) "d"x`
If `int_1^"a" (3x^2 + 2x + 1) "d"x` = 11, find the real value of a
`int_(-2)^2 sqrt((2 - x)/(2 + x))` = ?
`int_0^1 tan^-1 ((2x - 1)/(1 + x - x^2))` dx = ?
`int_2^3 "x"/("x"^2 - 1)` dx = ____________.
Evaluate the following definite integral:
`int_4^9 1/sqrtx dx`
Evaluate:
`int_(-π/2)^(π/2) (sin^3x)/(1 + cos^2x)dx`
Evaluate the following definite intergral:
`int_1^2 (3x)/((9x^2-1 )`dx
Evaluate the following definite integral:
`int_1^2 (3x)/((9x^2 - 1)) dx`
Evaluate the following definite intergral:
`int_1^3logxdx`
Evaluate the following definite intergral:
`int_1^2 (3x)/ ((9x^2 -1)) dx`
Evaluate the following definite intergral:
`int_4^9(1)/sqrtxdx`
Evaluate the following definite integrals: `int_4^9 (1)/sqrt(x)*dx`
Evaluate the following definite integrals: `int_1^2 (3x)/((9x^2 - 1))*dx`
Solve the following.
`int_1^3x^2 logx dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
Evaluate the integral.
`int_-9^9 x^3/(4-x^2) dx`
Solve the following.
`int_0^1e^(x^2) x^3 dx`
Evaluate the following definite integral:
`int_-2^3 1/(x+5).dx`
Evaluate the following definite intergral:
`int_(-2)^3 1/(x + 5)dx`
