Advertisements
Advertisements
प्रश्न
Evaluate `int_1^3 x^2*log x "d"x`
उत्तर
Let I = `int_1^3 x^2*log x "d"x`
= `[log x int x^2 "d"x]_1^3 - int_1^3["d"/("d"x)(log x) intx^2 "d"x]"d"x`
= `[log x* x^3/3]_1^3 - int_1^3 1/x*x^3/3 "d"x`
= `[9log3 - log1*1/3] - 1/3 int_1^3 x^2 "d"x`
= `(9log 3 - 0) - 1/3 [x^3/3]_1^3`
= `9log3 - 1/3(27/3 - 1/3)`
= `9log3 - 1/3(26/3)`
∴ I = `9log 3 - 26/9`
APPEARS IN
संबंधित प्रश्न
Evaluate : `intlogx/(1+logx)^2dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^1 x(1-x)^n dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^2 xsqrt(2 -x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_((-pi)/2)^(pi/2) sin^2 x dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi (x dx)/(1+ sin x)`
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (sin x - cos x)/(1+sinx cos x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi log(1+ cos x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^a sqrtx/(sqrtx + sqrt(a-x)) dx`
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.
The value of `int_0^(pi/2) log ((4+ 3sinx)/(4+3cosx))` dx is ______.
Evaluate : `int "e"^(3"x")/("e"^(3"x") + 1)` dx
Evaluate the following integral:
`int_0^1 x(1 - x)^5 *dx`
`int_0^(pi/4) (sec^2 x)/((1 + tan x)(2 + tan x))`dx = ?
`int_2^3 x/(x^2 - 1)` dx = ______
`int_0^(pi/2) sqrt(cos theta) * sin^2 theta "d" theta` = ______.
`int_-2^1 dx/(x^2 + 4x + 13)` = ______
`int_0^pi x sin^2x dx` = ______
Evaluate `int_(-1)^2 "f"(x) "d"x`, where f(x) = |x + 1| + |x| + |x – 1|
`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.
`int_0^(pi/2) sqrt(1 - sin2x) "d"x` is equal to ______.
`int_0^(2"a") "f"("x") "dx" = int_0^"a" "f"("x") "dx" + int_0^"a" "f"("k" - "x") "dx"`, then the value of k is:
The value of `int_0^1 tan^-1 ((2x - 1)/(1 + x - x^2)) dx` is
Evaluate: `int_0^(π/2) 1/(1 + (tanx)^(2/3)) dx`
Evaluate: `int_0^(2π) (1)/(1 + e^(sin 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`
If `intxf(x)dx = (f(x))/2` then f(x) = ex.
`int_0^1|3x - 1|dx` equals ______.
If `β + 2int_0^1x^2e^(-x^2)dx = int_0^1e^(-x^2)dx`, then the value of β is ______.
If f(x) = `{{:(x^2",", "where" 0 ≤ x < 1),(sqrt(x)",", "when" 1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals ______.
What is `int_0^(π/2)` sin 2x ℓ n (cot x) dx equal to ?
With the usual notation `int_1^2 ([x^2] - [x]^2)dx` is equal to ______.
If `int_0^(π/2) log cos x dx = π/2 log(1/2)`, then `int_0^(π/2) log sec dx` = ______.
Evaluate: `int_1^3 sqrt(x + 5)/(sqrt(x + 5) + sqrt(9 - x))dx`
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
Evaluate `int_1^2(x+3)/(x(x+2)) dx`
Evaluate the following integral:
`int_-9^9 x^3 / (4 - x^2) dx`
Solve the following.
`int_2^3x/((x+2)(x+3))dx`
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
