Advertisements
Advertisements
Question
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
Advertisements
Solution
Let f (x) = cos5 x
Now we have
f (2π - x) = (cos (2π - x))5
= (cos x)5 = cos5 x = f (x)
⇒ `I = 2 int_0^pi cos^5 x dx`
`[∵ int_0^(2a) f (x) dx = 2 int_0^a f (x)dx, if (2a - x) = f(x) = 0, if (2a - x) = -f(x)]`
Again, we have
f (π - x) = (cos (π - x))5 = -cos5 x = - f(x)
⇒ `2 int_0^pi cos^5 x dx = 0`
Hence, `int_0^(2pi) cos^5 x dx `
`= 2 int_0^5 cos^5 x dx `
= 2 × 0
= 0
APPEARS IN
RELATED QUESTIONS
Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`
Evaluate : `intlogx/(1+logx)^2dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^1 x(1-x)^n dx`
`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.
\[\int\limits_0^a 3 x^2 dx = 8,\] find the value of a.
Evaluate : `int "e"^(3"x")/("e"^(3"x") + 1)` dx
`int_"a"^"b" "f"(x) "d"x` = ______
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x)) dx` = ______.
Choose the correct alternative:
`int_(-9)^9 x^3/(4 - x^2) "d"x` =
State whether the following statement is True or False:
`int_(-5)^5 x/(x^2 + 7) "d"x` = 10
Evaluate `int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x)) "d"x`
`int_-9^9 x^3/(4 - x^2)` dx = ______
`int_0^{pi/2} xsinx dx` = ______
The value of `int_1^3 dx/(x(1 + x^2))` is ______
`int_0^1 "dx"/(sqrt(1 + x) - sqrtx)` = ?
`int_(-1)^1 log ((2 - x)/(2 + x)) "dx" = ?`
Find `int_0^(pi/4) sqrt(1 + sin 2x) "d"x`
`int_("a" + "c")^("b" + "c") "f"(x) "d"x` is equal to ______.
Evaluate the following:
`int_(-pi/4)^(pi/4) log|sinx + cosx|"d"x`
`int_0^(pi/2) cos x "e"^(sinx) "d"x` is equal to ______.
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
Evaluate: `int_((-π)/2)^(π/2) (sin|x| + cos|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_a^b f(x)dx = int_a^b f(x - a - b)dx`.
`int_0^1|3x - 1|dx` equals ______.
`int_0^(π/4) x. sec^2 x dx` = ______.
`int_0^(π/2)((root(n)(secx))/(root(n)(secx + root(n)("cosec" x))))dx` is equal to ______.
`int_-1^1 (17x^5 - x^4 + 29x^3 - 31x + 1)/(x^2 + 1) dx` is equal to ______.
Assertion (A): `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x))dx` = 3.
Reason (R): `int_a^b f(x) dx = int_a^b f(a + b - x) dx`.
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is ______.
Solve the following.
`int_1^3 x^2 logx dx`
If `int_0^1(3x^2 + 2x+a)dx = 0,` then a = ______
Evaluate:
`int_0^1 |2x + 1|dx`
Solve.
`int_0^1e^(x^2)x^3dx`
Evaluate the following definite intergral:
`int_1^2 (3x)/(9x^2 - 1) dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
Evaluate:
`int_0^sqrt(2)[x^2]dx`
Evaluate the following definite intergral:
`int_1^3logx dx`
