AntiderivativeDetermine the antiderivative of the integral (2x+5)e^(x^2+5x).
You need to perform the inverse operation to differentiation, hence, performing integration yields:
`int (2x + 5)e^(x^2 + 5x) dx `
You should come up with the substitution, such that:
`x^2 + 5x = u => (2x + 5)dx = du`
Using the variable u, yields:
`int e^u*du = e^u + c`
Replacing back `x^2 + 5x` for u yields:
`int (2x + 5)e^(x^2 + 5x) dx = e^(x^2 + 5x) + c`
Hence, evaluating the antiderivative of the given function, using the replacement of original variable, yields `int (2x + 5)e^(x^2 + 5x) dx = e^(x^2 + 5x) + c.`
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