To determine the antiderivative, we'll have to compute the indefinite integral of the function f(x) = (2x+5)*e^(x^2+5x)
Int (2x+5)*e^(x^2+5x) dx
We notice that the exponent of e is a function whose derivative is the other factor of the integrand.
We'll note the exponent by t = x^2+5x and we'll solve the integral using substitution method.
t = x^2+5x
We'll differentiate both sides:
dt = (x^2+5x)'dx
dt = (2x + 5)dx
Now, we'll re-write the integral changing the variable:
Int (2x+5)*e^(x^2+5x) dx = Int e^t dt
Int e^t dt = e^t + C
But t = x^2+5x
Int (2x+5)*e^(x^2+5x) dx = e^(x^2+5x) + C
To find the antiderivative of 2x+5)(e^x^2+5x).
Let f(x) = (2x+5)e^(x^2+5x).
We have to find the Inegral (or anti derivative of 2x+5)e^(x^2+5x) dx...(1)
We put x^2+5x = t
Differentiating x^2+5x = t, we get:
(2x+5)dx = dt.
Therefore substituting t = x^2+5x = t and (2x+5) dx = t in 2x+5) e^(x^2+5x) dx, we get:
Integral f(x)dx = Integral e^t dt = e^t + C, where C is the constant of integration.
Integral f(x) dx = e^(x^2+5x) + C is the anti derivatve of (2x^2+5)e^(x^2+5x).