Advertisements
Advertisements
Question
Evaluate the following : `int_0^1 t^2 sqrt(1 - t)*dt`
Solution
We use the property,
`int_0^a f(t)*dt = int_0^a f(a - t)*dt`
∴ `int_0^1 t^2 sqrtt(1 - t)*dt = int_0^1 (1 - t)^2 sqrt(1 - 1 + t)*dt`
= `int_0^1 (1 - 2t + t^2)sqrt(t)*dt`
= `int_0^1 (t^(1/2) - 2t^(3/2) + t^(5/2))*dt`
= `[(t^(3/2))/(3/2) - 2*(t(5)/(2))/(5/2) + (t^(7/2))/(7/2)]_0^1`
= `(2)/(3)(1)^(3/2) - (4)/(5)(1)^(5/2) + (2)/(7)(1)^(7/2) - 0`
= `(2)/(3) - (4)/(5) + (2)/(7) - 0`
= `(70 - 84 + 30)/(105)`
= `(16)/(105)`.
APPEARS IN
RELATED QUESTIONS
Evaluate : `int_0^(pi/4) sin^4x*dx`
Evaluate : `int_0^1 x tan^-1x*dx`
Evaluate : `int_0^(pi/4) (sec^2x)/(3tan^2x + 4tan x +1)*dx`
Evaluate the following : `int_0^3 x^2(3 - x)^(5/2)*dx`
Evaluate the following:
`int_0^pi x/(1 + sin^2x) * dx`
Evaluate the following : `int_(pi/5)^((3pi)/10) sinx/(sinx + cosx)*dx`
Evaluate the following definite integrals: If `int_0^"a" (2x + 1)*dx` = 2, find the real value of a.
Evaluate the following integrals : `int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx`
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_(-9)^9 x^3/(4 - x^2)*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_3^5 dx/(sqrt(x + 4) + sqrt(x - 2)`
Prove that: `int_0^"a" "f"(x) "d"x = int_0^"a" "f"("a" - x) "d"x`. Hence find `int_0^(pi/2) sin^2x "d"x`
`int_1^2 x^2 "d"x` = ______
Evaluate `int_0^1 1/(sqrt(1 + x) + sqrt(x)) "d"x`
Evaluate `int_2^3 x/((x + 2)(x + 3)) "d"x`
Evaluate `int_1^3 log x "d"x`
Evaluate the following definite integrats:
`int_4^9 1/sqrt x dx`
Evaluate the following definite intergral:
`int_4^9 1/sqrt(x)dx`
Evaluate the following definite integral :
`int_1^2 (3"x")/((9"x"^2 - 1)) "dx"`
Evaluate the following integrals:
`int_0^1 x(1 - x)^5 dx`
Solve the following.
`int_1^3 x^2 logx dx`
Evaluate the following definite integral:
`int_4^9 1/sqrtx dx`
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 _1^3logxdx`
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
Prove that `int_0^(2a) f(x)dx = int_0^a[f(x) + f(2a - x)]dx`
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
Evaluate the following definite integral:
`int_4^9 1/sqrtx dx`
Evaluate the following definite intergral:
`int_-2^3 1/(x+5)dx`
Solve the following.
`int_0 ^1 e^(x^2) * x^3`dx
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 intergral:
`int_1^2(3x)/(9x^2-1).dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
Solve the following.
`int_1^3x^2 logx dx`
