Integrate sinx^2.xdx???
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integration calculator online emathhelp
when you see list of results click on :
Integral Calculator with Steps - eMathHelp
When page be open In rectangle type :
sinx^2.xdx then clic button CALCULATE
You will se solution step - by -step
we know that
cos 2x = 1 - 2sin^2 x
2sin^2 x = 1 - cos 2x
sin^2 x = 1/2 - (1/2)cos(2x)
so integral(sin^2 x)dx
= integral( 1/2 - (1/2)cos(2x) ) dx
= (1/2)x - (1/4)sin(2x) + costant
To make things a bit clearer,
∫sin(x^2) x dx
let
u = x^2
du = 2x dx
and you have
∫1/2 sin(u) du
not so hard, eh?
To integrate the function ∫sin^2(x) * x dx, we can use integration by parts.
Integration by parts is based on the product rule for differentiation, which states that the derivative of the product of two functions, u(x) and v(x), is given by (u*v)' = u'v + uv'.
The formula for integration by parts is:
∫ u(x) * v'(x) dx = u(x)v(x) - ∫ v(x) * u'(x) dx
In our case, let's choose u(x) = x and dv(x) = sin^2(x) dx. We need to find du(x) and v(x).
Differentiating u(x) = x with respect to x:
du(x)/dx = 1
To find v(x), we integrate dv(x) = sin^2(x) dx. This can be done using a trigonometric identity. The identity that will help us is:
sin^2(x) = (1 - cos(2x)) / 2
Substituting this identity into our integral, we get:
∫ sin^2(x) dx = ∫ (1 - cos(2x))/2 dx
Now, let's solve this integral:
∫ (1 - cos(2x))/2 dx = (1/2) * ∫ (1 - cos(2x)) dx
= (1/2) * [∫ dx - ∫ cos(2x) dx]
= (1/2) * [x - (1/2) * sin(2x) + C]
where C is the constant of integration.
Now, substituting back u(x) = x and v(x) = (1/2) * (x - (1/2) * sin(2x)), we can apply the integration by parts formula:
∫ sin^2(x) * x dx = u(x) * v(x) - ∫ v(x) * u'(x) dx
= x * (1/2) * [x - (1/2) * sin(2x)] - ∫ [(1/2) * [x - (1/2) * sin(2x)]] dx
= (1/2) * [x^2 - (1/2) * x * sin(2x)] - (1/4) * ∫ x dx + (1/4) * ∫ sin(2x) dx
= (1/2) * [x^2 - (1/2) * x * sin(2x)] - (1/4) * [x^2 / 2] - (1/4) * [-1/2 * cos(2x)]
= (1/2) * [x^2 - (1/2) * x * sin(2x)] - (1/8) * x^2 + (1/8) * cos(2x) + C
So the final result of integrating sin^2(x) * x dx is:
(1/2) * [x^2 - (1/2) * x * sin(2x)] - (1/8) * x^2 + (1/8) * cos(2x) + C
where C is the constant of integration.