Advertisements
Advertisements
प्रश्न
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is ______.
विकल्प
–1
0
1
2
Advertisements
उत्तर
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is 0.
Explanation:
f(x) = `e^(cos^2x) sin^3 (2n + 1)x`
f(–x) = `e^(cos^2(-x)) sin^3 (2n + 1)(-x)`
f(–x) = `-e^(cos^2x) sin^3 (2n + 1)x`
∵ f(–x) = –f(x)
So, `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` = 0
APPEARS IN
संबंधित प्रश्न
Show that `int_0^a f(x)g (x)dx = 2 int_0^a f(x) dx` if f and g are defined as f(x) = f(a-x) and g(x) + g(a-x) = 4.
Evaluate : ∫ log (1 + x2) dx
Evaluate = `int (tan x)/(sec x + tan x)` . dx
Prove that `int_0^"a" "f" ("x") "dx" = int_0^"a" "f" ("a" - "x") "d x",` hence evaluate `int_0^pi ("x" sin "x")/(1 + cos^2 "x") "dx"`
`int_-9^9 x^3/(4 - x^2)` dx = ______
`int_0^{pi/2} xsinx dx` = ______
`int_"a"^"b" sqrtx/(sqrtx + sqrt("a" + "b" - x)) "dx"` = ______.
`int_0^1 x tan^-1x dx` = ______
The value of `int_1^3 dx/(x(1 + x^2))` is ______
`int_0^pi sin^2x.cos^2x dx` = ______
`int_0^9 1/(1 + sqrtx)` dx = ______
`int_0^(pi/2) 1/(1 + cos^3x) "d"x` = ______.
`int_0^(pi/2) cos x "e"^(sinx) "d"x` is equal to ______.
Evaluate: `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`
Evaluate: `int_0^(π/2) 1/(1 + (tanx)^(2/3)) dx`
Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
Evaluate: `int_(-1)^3 |x^3 - x|dx`
If `int_a^b x^3 dx` = 0, then `(x^4/square)_a^b` = 0
⇒ `1/4 (square - square)` = 0
⇒ b4 – `square` = 0
⇒ (b2 – a2)(`square` + `square`) = 0
⇒ b2 – `square` = 0 as a2 + b2 ≠ 0
⇒ b = ± `square`
`int_4^9 1/sqrt(x)dx` = ______.
If `int_0^1(sqrt(2x) - sqrt(2x - x^2))dx = int_0^1(1 - sqrt(1 - y^2) - y^2/2)dy + int_1^2(2 - y^2/2)dy` + I then I equal.
`int_0^1|3x - 1|dx` equals ______.
`int_0^π(xsinx)/(1 + cos^2x)dx` equals ______.
Evaluate `int_0^(π//4) log (1 + tanx)dx`.
Solve the following.
`int_1^3 x^2 logx dx`
Evaluate the following definite integral:
`int_1^3 log x dx`
Solve the following.
`int_0^1e^(x^2)x^3 dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
