Advertisements
Advertisements
Question
Evaluate the following : `int_((-pi)/4)^(pi/4) x^3 sin^4x*dx`
Solution
Let I = `int_((-pi)/4)^(pi/4) x^3 sin^4x*dx`
Let f(x) = `x^3 sin^4x`
∴ f( –x) = `(-x)^3 sin^4(- x)`
= `-x^3sin^4x`
= `-f(x)`
∴ f is an odd function.
∴ `int_((-pi)/4)^(pi/4) f(x)*dx = 0, "i.e." int_((-pi)/4)^(pi/4) x^3 sin^4x*dx` = 0.
APPEARS IN
RELATED QUESTIONS
Prove that:
`int 1/(a^2 - x^2) dx = 1/2 a log ((a +x)/(a-x)) + c`
Evaluate : `int_2^3 (1)/(x^2 + 5x + 6)*dx`
Evaluate : `int_0^(pi/2) cosx/((1 + sinx)(2 + sin x))*dx`
Evaluate : `int_(-1)^1 (1)/(a^2e^x + b^2e^(-x))*dx`
Evaluate : `int_0^(pi/4) sec^4x*dx`
Choose the correct option from the given alternatives :
`int_0^(pi/2) sn^6x cos^2x*dx` =
Choose the correct option from the given alternatives :
The value of `int_((-pi)/4)^(pi/4) log((2+ sin theta)/(2 - sin theta))*d theta` is
Evaluate the following : `int_0^1 t^5 sqrt(1 - t^2)*dt`
Evaluate the following : `int_1^oo 1/(sqrt(x)(1 + x))*dx`
Evaluate the following : `int_0^(pi/2) [2 log (sinx) - log (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 definite integrals: `int_0^1 (1)/(sqrt(1 + x) + sqrt(x))*dx`
Evaluate the following integrals : `int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx`
Fill in the blank : `int_0^2 e^x*dx` = ________
Fill in the blank : `int_0^1 dx/(2x + 5)` = _______
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(x - "a" - "b")*dx`
Solve the following : `int_2^3 x/((x + 2)(x + 3))*dx`
Solve the following : `int_1^2 (x + 3)/(x (x + 2))*dx`
Solve the following : `int_1^2 e^(2x) (1/x - 1/(2x^2))*dx`
Solve the following : `int_3^5 dx/(sqrt(x + 4) + sqrt(x - 2)`
Solve the following : `int_1^2 (5x^2)/(x^2 + 4x + 3)*dx`
Solve the following : `int_1^2 dx/(x(1 + logx)^2`
Prove that: `int_"a"^"b" "f"(x) "d"x = int_"a"^"c""f"(x) "d"x + int_"c"^"b" "f"(x) "d"x`, where a < c < b
`int_0^"a" 4x^3 "d"x` = 81, then a = ______
State whether the following statement is True or False:
`int_"a"^"b" "f"(x) "d"x = int_"a"^"b" "f"("a" + "b" - x) "d"x`
State whether the following statement is True or False:
`int_0^(2"a") "f"(x) "d"x = int_0^"a" "f"(x) "d"x + int_0^"a" "f"("a" - x) "d"x`
If `int_0^"a" (2x + 1) "d"x` = 2, find a
Evaluate `int_0^"a" x^2 ("a" - x)^(3/2) "d"x`
Evaluate the following definite integrats:
`int_4^9 1/sqrt x dx`
Evaluate the following definite integral:
`int_1^3 log x dx`
`int_0^1 1/(2x + 5)dx` = ______
Solve the following.
`int_0^1 e^(x^2) x^3 dx`
Evaluate the following definite integrals: `int_4^9 (1)/sqrt(x)*dx`
Evaluate the following definite intergral:
`int_1^3 log x dx`
Evaluate the following definite intergral:
`int_-2^3 1/(x+5)dx`
Evaluate the following definite intergral:
`int_1^3 log x dx`
Evaluate the following definite intergral:
`int_(1)^3logx dx`
