AntiderivativeDetermine the antiderivative of  the integral (2x+5)e^(x^2+5x).

Asked on by vladianna

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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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.`

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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.

We have:

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

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