Advertisements
Advertisements
Question
Evaluate:
`int_0^(pi/2) sqrt(cos x) sin^3x * dx`
Solution
Let I = `int_0^(pi/2) sqrt(cos x) sin^3x * dx`
= `int_0^(pi/2) sqrt(cosx)sin^2x sinx * dx`
= `int_0^(pi/2) sqrt(cosx)(1 - cos^2x)sinx*dx`
Put cos x = t
∴ – sin x · dx = dt
∴ sin x · dx = – dt
When x = 0, t = cos 0 = 1
When x = π/2, t = cos 2π = 0
`I = - int_1^0 sqrt(t)(1 - t^2)(dt)`
`I = - int_1^0 (t^(1//2) - t^(5//2)) dt`
`I = -[(2t)^(3//2)/3 - (2t)^(7//2)/7]_1^0`
`I = -[0 - (2/3 - 2/7)]`
`I = (14 - 6)/21`
`I = 8/21`
APPEARS IN
RELATED QUESTIONS
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_0^4 (1)/sqrt(4x - x^2)*dx`
Evaluate : `int_(-1)^1 (1)/(a^2e^x + b^2e^(-x))*dx`
Evaluate: `int_0^(pi/2) sin2x*tan^-1 (sinx)*dx`
Evaluate the following:
`int_0^(pi/2) log(tanx)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_0^9 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx` =
Evaluate the following : `int_(pi/4)^(pi/2) (cos theta)/[cos theta/2 + sin theta/2]^3*d theta`
Evaluate the following : `int_0^1 (cos^-1 x^2)*dx`
Evaluate the following:
`int_0^pi x/(1 + sin^2x) * dx`
Evaluate the following : `int_0^1 sin^-1 ((2x)/(1 + x^2))*dx`
Evaluate the following : `int_0^4 [sqrt(x^2 + 2x + 3]]^-1*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 : If `int_0^k 1/(2 + 8x^2)*dx = pi/(16)`, find k
Evaluate the following definite integral:
`int_(-2)^3 (1)/(x + 5)*dx`
Evaluate the following integrals : `int_(-9)^9 x^3/(4 - x^2).dx`
Choose the correct alternative :
`int_(-9)^9 x^3/(4 - x^2)*dx` =
State whether the following is True or False : `int_(-5)^(5) x^3/(x^2 + 7)*dx` = 0
State whether the following is True or False : `int_4^7 ((11 - x)^2)/((11 - x)^2 + x^2)*dx = (3)/(2)`
Solve the following : `int_0^9 (1)/(1 + sqrt(x))*dx`
State whether the following statement is True or False:
`int_0^1 1/(2x + 5) "d"x = log(7/5)`
If `int_0^"a" (2x + 1) "d"x` = 2, find a
Evaluate `int_0^"a" x^2 ("a" - x)^(3/2) "d"x`
`int_(-2)^2 sqrt((2 - x)/(2 + x))` = ?
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 integrats:
`int_4^9 1/sqrt x dx`
Evaluate the following definite integrals:
`int _1^2 (3x) / ( (9 x^2 - 1)) * dx`
Evaluate the following definite intergral:
`int_1^3 log xdx`
`int_0^1 1/(2x + 5)dx` = ______
Evaluate the following definite integral:
`int_1^2 (3x)/((9x^2 - 1))dx`
Evaluate the following definite intergral:
`int_1^2 (3x)/((9x^2-1 )`dx
Prove that `int_0^(2a) f(x)dx = int_0^a[f(x) + f(2a - x)]dx`
If `int_((-pi)/4) ^(pi/4) x^3 * sin^4 x dx` = k then k = ______.
Evaluate the following definite integral:
`int_1^2 (3x)/((9x^2 - 1)) dx`
Evaluate the following definite integral:
`int_1^3 logx dx`
Evaluate the following definite intergral:
`int_4^9 1/sqrtxdx`
Solve the following.
`int_1^3x^2logx dx`
Evaluate the following definite intergral:
`int_(1)^3logx dx`
