Advertisements
Advertisements
प्रश्न
Evaluate the following : `int_0^pi x*sinx*cos^4x*dx`
उत्तर
Let I = `int_0^pi x*sinx*cos^4x*dx` ...(1)
We use the property, `int_0^a f(x)*dx = int_0^a f(a - x)*dx`
Here a = π.
Hence changing x by π – x, we get
I = `int_0^pi (pi - x)*sin(pi - x)*[cos(pi - x)]^4*dx`
= `int_0^pi (pi - x)*sinx*cos^4x*dx` ...(2)
Adding(1) and (2), we get
2I = `int_0^pi x*sinx*cos^4x*dx + int_0^pi (pi - x)*sinx*cos^4x*dx`
= `int_0^pi (x + pi - x)*sinx*cos^4x*dx`
= `pi int_0^pi sinx*cos^4x*dx`
∴ I = `pi/(2) int_0^pi cos^4x*sinx*dx`
Put cos = t
∴ – sinx · dx = dt
∴ sinx · dx = – dt
When x 0, t = cos 0 = 1
When x = π cos π = – 1
∴ I = `pi/(2) int_1^(-1) t^4(- dt)`
= `- pi/(2) int_(1)^(-1) t^4*dt`
= `- pi/(2)[(t^5)/5]_1^(-1)`
= `- pi/(10)[t^5]_1^(-1)`
= `- pi/(10)[(- 1)^5 - (1)^5]`
= `- pi/(10) (- 1 - 1)`
= `(2pi)/(10)`
= `pi/(5)`.
APPEARS IN
संबंधित प्रश्न
Evaluate:
`int_0^1 (1)/sqrt(3 + 2x - x^2)*dx`
Evaluate : `int_0^(1/sqrt(2)) (sin^-1x)/(1 - x^2)^(3/2)*dx`
Evaluate : `int_0^(pi/2) (sinx - cosx)/(1 + sinx cosx)*dx`
Evaluate the following : `int_((-pi)/4)^(pi/4) (x + pi/4)/(2 - cos 2x)*dx`
Evaluate the following : `int_(-a)^(a) (x + x^3)/(16 - x^2)*dx`
`int_0^(log5) (e^x sqrt(e^x - 1))/(e^x + 3) * dx` = ______.
Evaluate the following : `int_0^1 1/(1 + sqrt(x))*dx`
Evaluate the following : `int_(-1)^(1) (1 + x^3)/(9 - x^2)*dx`
Evaluate the following : If `int_0^k 1/(2 + 8x^2)*dx = pi/(16)`, find k
Evaluate the following definite integral:
`int_1^3 logx.dx`
Evaluate the following integrals:
`int_1^3 (root(3)(x + 5))/(root(3)(x + 5) + root(3)(9 - x))*dx`
Fill in the blank : `int_2^3 x^4*dx` = _______
Fill in the blank : If `int_0^"a" 3x^2*dx` = 8, then a = _______
State whether the following is True or False : `int_1^2 sqrt(x)/(sqrt(3 - x) + sqrt(x))*dx = (1)/(2)`
Solve the following : `int_2^4 x/(x^2 + 1)*dx`
Solve the following : `int_1^2 dx/(x(1 + logx)^2`
If `int_1^"a" (3x^2 + 2x + 1) "d"x` = 11, find the real value of a
`int_((-pi)/8)^(pi/8) log ((2 - sin x)/(2 + sin x))` dx = ______.
Prove that: `int_0^(2a) f(x)dx = int_0^a f(x)dx + int_0^a f(2a - x)dx`
Evaluate the following definite integrals: `int_-2^3 1/(x + 5) *dx`
Evaluate the following integral:
`int_0^1 x(1-x)^5dx`
Solve the following.
`int_1^3x^2 logx dx`
`int_0^1 1/(2x + 5)dx` = ______
Evaluate the following definite integral:
`int_1^2 (3x)/((9x^2 - 1))dx`
`int_0^4 1/sqrt(4x - x^2)dx` = ______.
Evaluate the following definite intergral:
`int_4^9 1/sqrt(x)dx`
Evaluate the following definite intergral:
`int_1^2 (3x)/((9x^2-1 )`dx
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
Evaluate the following definite integral:
`int_1^2 (3x)/((9x^2 - 1)) dx`
Evaluate the following definite integral:
`int_4^9 1/sqrtx dx`
Evaluate the following definite intergral:
`int_4^9 1/sqrtxdx`
Evaluate the following definite intergral:
`int_-2^3 1/(x+5)dx`
Evaluate the following definite intergral:
`int_1^2(3x)/(9x^2-1).dx`
Evaluate the following definite intergral:
`int_-2^3 1/(x+5).dx`
Solve the following.
`int_1^3 x^2 log x 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_4^9 1/sqrtx dx`
