Advertisements
Advertisements
प्रश्न
Evaluate the following : `int_(-a)^(a) (x + x^3)/(16 - x^2)*dx`
उत्तर
Let I = `int_(-a)^(a) (x + x^3)/(16 - x^2)*dx`
Let f(x) = `(x + x^3)/(16 - x^2)`
∴ f( –x) = `((-x)+(-x)^3)/(16 - (- x)^2`
= `(-(x+x^3))/(16 - x^2)`
= `-f(x)`
∴ f is an odd function.
∴ `int_-a^a f(x)*dx = 0, "i.e." int_a^a (x + x^3)/(16 - x^2)*dx` = 0.
APPEARS IN
संबंधित प्रश्न
Show that: `int _0^(pi/4) log (1 + tanx) dx = pi/8 log2`
Evaluate : `int_0^(1/sqrt(2)) (sin^-1x)/(1 - x^2)^(3/2)*dx`
Evaluate:
`int_0^(pi/2) sqrt(cos x) sin^3x * dx`
Evaluate : `int_0^pi (1)/(3 + 2sinx + cosx)*dx`
Evaluate the following : `int_(-3)^(3) x^3/(9 - x^2)*dx`
Evaluate the following : `int_0^1 t^2 sqrt(1 - t)*dt`
Choose the correct option from the given alternatives :
If `[1/logx - 1/(logx)^2]*dx = a + b/(log2)`, then
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 t^5 sqrt(1 - t^2)*dt`
Evaluate the following:
`int_0^pi x/(1 + sin^2x) * dx`
Evaluate the following : `int_0^(pi/2) [2 log (sinx) - log (sin 2x)]*dx`
Evaluate the following : `int_0^4 [sqrt(x^2 + 2x + 3]]^-1*dx`
Evaluate the following definite integrals: If `int_0^"a" (2x + 1)*dx` = 2, find the real value of a.
Evaluate the following definite integrals: if `int_1^"a" (3x^2 + 2x + 1)*dx` = 11, find a.
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` =
Choose the correct alternative :
If `int_0^"a" 3x^2*dx` = 8, then a = ?
Fill in the blank : `int_2^3 x^4*dx` = _______
State whether the following is True or False : `int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx = (9)/(2)`
Solve the following : `int_1^2 x^2*dx`
Solve the following : `int_0^1 (1)/(sqrt(1 + x) + sqrt(x))dx`
Solve the following : `int_0^4 (1)/sqrt(x^2 + 2x + 3)*dx`
Solve the following : `int_1^2 (5x^2)/(x^2 + 4x + 3)*dx`
`int_1^9 (x + 1)/sqrt(x) "d"x` =
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`
Choose the correct alternative:
`int_0^"a" 3x^5 "d"x` = 8, then a =
Choose the correct alternative:
`int_(-2)^3 1/(x + 5) "d"x` =
State whether the following statement is True or False:
`int_2^3 x/(x^2 + 1) "d"x = 1/2 log 2`
Evaluate `int_0^1 1/(sqrt(1 + x) + sqrt(x)) "d"x`
If `int_1^"a" (3x^2 + 2x + 1) "d"x` = 11, find the real value of a
Evaluate `int_1^3 log x "d"x`
`int_0^1 tan^-1 ((2x - 1)/(1 + x - x^2))` dx = ?
Evaluate the following definite integrals: `int_-2^3 1/(x + 5) *dx`
Evaluate the following definite integral :
`int_1^2 (3"x")/((9"x"^2 - 1)) "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 integral:
`int_0^1x(1-x)^5dx`
Evaluate the following definite integral:
`int_-2^3 1/(x+5).dx`
