What is the antiderivative of  integrand (2x+5)e^(x^2+5x)?

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giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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

neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

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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).

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