Advertisements
Advertisements
प्रश्न
Evaluate the following : `int_0^pi (sin^-1x + cos^-1x)^3 sin^3x*dx`
उत्तर
Let I = `int_0^pi (sin^-1x + cos^-1x)^3 sin^3x*dx`
We know that, sin–1x + cos–1x = `pi/(2)`
and
sin 3x = 3 sin x – 4 sin3x
∴ 4sin3x = 3 sin x – sin 3x
∴ sin3x = `3/4 sinx - 1/4 sin3x`
∴ I = `int_0^pi (pi/2)^3[3/4 sin x - 1/4 sin 3x]*dx`
= `pi^3/(8) xx 3/4 int_0^pi sin x*dx - pi^2/(8) xx 1/4 int_0^pi sin3x`
= `(3pi^3)/(32) [- cos pi - ( - cos 0)] - pi^3/(32)[- (cos 3pi)/(3) - ((- cos0)/3)]`
= `(3pi^3)/(32)[1 + 1] - pi^3/(32)[1/3 + 1/3]`
= `(6pi^3)/(32) - (2pi^3)/(96)`
= `(18pi^3 - 2pi^3)/(96)`
= `(16pi^3)/(96)`
= `pi^3/(6)`.
APPEARS IN
संबंधित प्रश्न
Prove that:
`{:(int_(-a)^a f(x) dx = 2 int_0^a f(x) dx",", "If" f(x) "is an even function"),( = 0",", "if" f(x) "is an odd function"):}`
Evaluate:
`int_0^(pi/4) sqrt(1 + sin 2x)*dx`
Evaluate : `int_0^4 (1)/sqrt(4x - x^2)*dx`
Evaluate : `int_0^(pi//4) (sin2x)/(sin^4x + cos^4x)*dx`
Evaluate : `int_0^(pi/2) cosx/((1 + sinx)(2 + sin x))*dx`
Evaluate: `int_0^(pi/2) sin2x*tan^-1 (sinx)*dx`
Evaluate the following : `int_(-3)^(3) x^3/(9 - x^2)*dx`
Evaluate the following:
`int_((-pi)/2)^(pi/2) log((2 + sin x)/(2 - sin x)) * dx`
Evaluate the following : `int_((-pi)/4)^(pi/4) (x + pi/4)/(2 - cos 2x)*dx`
Evaluate the following : `int_((-pi)/4)^(pi/4) x^3 sin^4x*dx`
Evaluate the following : `int_(-a)^(a) (x + x^3)/(16 - x^2)*dx`
Choose the correct option from the given alternatives :
`int_0^9 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx` =
Evaluate the following : `int_0^pi x*sinx*cos^4x*dx`
Evaluate the following : `int_0^(pi/2) [2 log (sinx) - log (sin 2x)]*dx`
Evaluate the following : If f(x) = a + bx + cx2, show that `int_0^1 f(x)*dx = (1/(6)[f(0) + 4f(1/2) + f(1)]`
Choose the correct alternative :
`int_2^3 x^4*dx` =
Choose the correct alternative :
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx` =
Fill in the blank : `int_4^9 (1)/sqrt(x)*dx` = _______
State whether the following is True or False : `int_(-5)^(5) x^3/(x^2 + 7)*dx` = 0
Solve the following : `int_(-4)^(-1) (1)/x*dx`
Choose the correct alternative:
`int_(-2)^3 1/(x + 5) "d"x` =
`int_1^2 x^2 "d"x` = ______
If `int_0^"a" (2x + 1) "d"x` = 2, find a
Evaluate `int_0^"a" x^2 ("a" - x)^(3/2) "d"x`
Evaluate the following integral:
`int_0^1 x(1-x)^5dx`
`int_0^4 1/sqrt(4x - x^2)dx` = ______.
Solve the following.
`int_1^3 x^2 log x dx`
Evaluate the following definite integral:
`int_-2^3 1/(x+5) *dx`
The principle solutions of the equation cos θ = `1/2` are ______.
Evaluate the following definite intergral:
`int_1^2(3x)/((9x^2-1))dx`
Evaluate the following definite intergral:
`int_4^9 1/sqrtx dx`
Evaluate the following definite intergral:
`int_1^3 log x·dx`
Evaluate the following definite intergral:
`int_-2^3 1/(x+5) · dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
