Advertisements
Advertisements
प्रश्न
Evaluate `int_0^"a" x^2 ("a" - x)^(3/2) "d"x`
उत्तर
Let I = `int_0^"a" x^2 ("a" - x)^(3/2) "d"x`
= `int_0^"a" ("a" - x)^2 ["a" - ("a" - x)]^(3/2) "d"x` ......`[because int_0^"a" "f"(x) "d"x = int_0^"a" "f"("a" - x) "d"x]`
= `int_0^"a"("a"^2 - 2"a"x + x^2)x^(3/2) "d"x`
= `int_0^"a"("a"^2x^(3/2) - 2"a"x^(5/2) + x^(7/2))"d"x`
= `"a"^2 int_0^"a" x^(3/2) "d"x - 2"a" int_0^"a" x^(5/2) "d"x + int_0^"a" x^(7/2) "d"x`
= `"a"^2[(x^(5/2))/(5/2)]_0^"a" - 2"a"[(x^(7/2))/(7/2)]_0^"a" + [(x^(9/2))/(9/2)]_0^"a"`
= `(2"a"^2)/5 [("a")^(5/2) - 0] - (4"a")/7 [("a")^(7/2) - 0] + 2/9 [("a")^(9/2) - 0]`
= `2/5"a"^(9/2) - 4/7"a"^(9/2) + 2/9"a"^(9/2)`
= `(2/5 - 4/7 + 2/9)"a"^(9/2)`
= `((126 - 180 + 70)/315)"a"^(9/2)`
∴ I = `16/315"a"^(9/2)`
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/2) sqrt(cos x) sin^3x * dx`
Evaluate the following : `int_(-3)^(3) x^3/(9 - x^2)*dx`
Evaluate the following : `int_(-1)^(1) (x^3 + 2)/sqrt(x^2 + 4)*dx`
Evaluate the following : `int_0^1 t^2 sqrt(1 - t)*dt`
Evaluate the following : `int_0^1 t^5 sqrt(1 - t^2)*dt`
Evaluate the following:
`int_0^pi x/(1 + sin^2x) * 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 f(x) = a + bx + cx2, show that `int_0^1 f(x)*dx = (1/(6)[f(0) + 4f(1/2) + f(1)]`
Evaluate the following definite integrals: `int_2^3 x/(x^2 - 1)*dx`
Evaluate the following definite integrals: `int_1^2 dx/(x^2 + 6x + 5)`
Evaluate the following integrals:
`int_1^3 (root(3)(x + 5))/(root(3)(x + 5) + root(3)(9 - x))*dx`
Choose the correct alternative :
`int_2^3 x/(x^2 - 1)*dx` =
Fill in the blank : `int_0^1 dx/(2x + 5)` = _______
Solve the following : `int_1^2 e^(2x) (1/x - 1/(2x^2))*dx`
Prove that: `int_"a"^"b" "f"(x) "d"x = int_"a"^"b" "f"("a" + "b" - x) "d"x`
Choose the correct alternative:
`int_2^3 x^4 "d"x` =
Evaluate:
`int_1^2 1/(x^2 + 6x + 5) dx`
Evaluate `int_1^2 (3x)/((9x^2 - 1)) "d"x`
Evaluate `int_0^1 "e"^(x^2)*"x"^3 "d"x`
`int_0^(pi/2) (cos x)/((4 + sin x)(3 + sin x))`dx = ?
`int_0^1 tan^-1 ((2x - 1)/(1 + x - x^2))` dx = ?
Evaluate the following definite intergral:
`int_1^3 log xdx`
Evaluate the following integral:
`int_0^1 x(1-x)^5dx`
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_1^2 (3x)/((9x^2 - 1)) dx`
Solve the following.
`int_0^1 e^(x^2) x^3 dx`
Evaluate the following definite intergral:
`int_-2^3 1/(x+5).dx`
Solve the following.
`int_1^3x^2 logx dx`
