evaluate the indefinite integral of (9xe^4x^2)dx
To evaluate the indefinite integral of 9xe^(4x^2) dx, we can use the technique of 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 is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function.
The formula for integration by parts is:
∫ u dv = uv - ∫ v du
In this case, we can assign u and dv as follows:
u = x (Note: u is the function that we differentiate)
dv = 9e^(4x^2) dx (Note: dv is the function that we integrate)
Next, we need to find du and v. Taking the derivative of u, we get:
du = 1 dx
To find v, we integrate dv:
v = ∫ 9e^(4x^2) dx
The integral of e^(4x^2) cannot be expressed in terms of elementary functions. Therefore, it is a special function known as the error function (erf). So, let's substitute this result into our formula for v.
v = ∫ 9e^(4x^2) dx = 9 ∫ e^(4x^2) dx = 9⋅erf(2x)
Now, we can substitute these values into the integration by parts formula:
∫ 9xe^(4x^2) dx = uv - ∫ v du
= x⋅(9⋅erf(2x)) - ∫ (9⋅erf(2x)) dx
Continuing, we can simplify the equation:
= 9x⋅erf(2x) - 9⋅∫ erf(2x) dx
The integral of the error function erf(2x) can be challenging to evaluate analytically. Therefore, this integral doesn't have a standard elementary function solution. However, if you need a numerical approximation of the integral, you can use numerical methods, such as Simpson's rule or the trapezoidal rule.
Hence, the indefinite integral of 9xe^(4x^2) dx is:
∫ 9xe^(4x^2) dx = 9x⋅erf(2x) - 9⋅∫ erf(2x) dx