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AntiderivativeDetermine the antiderivative of the integral (2x+5)e^(x^2+5x).
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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.`
Posted by sciencesolve on March 12, 2013 at 2:29 PM (Answer #3)
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
Posted by giorgiana1976 on May 30, 2011 at 5:27 PM (Answer #2)
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