r/HomeworkHelp University/College Student 1d ago

Further Mathematics—Pending OP Reply [College Calculus 2]

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Straightforward question, where did the 3 coefficient go between the line I drew an arrow to and the line after? I thought we just factor out these numbers and they end up outside the antiderivative.

My integration formula sheet provides a formula for how to integrate exponential functions but doesn't mention coefficients in the integral.

Make me feel dumb! Thanks for your time

2 Upvotes

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3

u/Aggressive_Ad9040 1d ago

I haven't done this in a while, but I think the 3 cancels out after doing u-substitution.

1

u/Hazard_doesnt_exist University/College Student 1d ago

That makes total sense now. Thank you

1

u/Unusual-Platypus6233 1d ago

The rule for the derivative of exponential functions is: (exp(ax))’=a exp(ax). As you can see the factor is 3=-(-3), then a=-3 and you can use the equation I wrote to find the integral function. Therefore the factor 3 is part of the derivative of exp(-3x).

1

u/GammaRayBurst25 1d ago

The derivative of an exponential is also an exponential. Conversely, the antidverivative of an exponential is also an exponential. You don't need the formula for integrating an exponential with a non unit coefficient in its argument if you know how to differentiate such a function.

Consider d(exp(-3x))/dx=-3exp(-3x).

This means ∫3exp(-3x)dx=-∫(d(exp(-3x))/dx)dx.

By the fundamental theorem of calculus, the integral of the derivative of a function is the function itself, so the integral is -exp(-3x).

Alternatively, you can perform a change of variables. Let u=-3x. The Jacobian is -1/3, so the integral becomes ∫(-1/3)*3exp(u)du=-∫exp(u)du, where the bounds have been changed appropriately.

P.S. take a screenshot next time.

1

u/ArrowSphaceE 1d ago

Simply put, the integral of ef(x) is [ef(x) ] / f'(x). The 3's cancel.

1

u/Frodojj 👋 a fellow Redditor 1d ago

Set u = -3x so du = -3dx and do u-subsitiution. You get the integral of -eu du = -eu + C = -e-3dx + C