What is the antiderivative of 3^x dx?

Question

I got 1/ln3 (3^x). Is this correct?

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Answer 1 · 2018-02-01T19:42:12

Yes, you are correct. For the information of others, an explanation is given.

Start with the given integral and then use the property of the natural logarithm and its inverse #u = e^ln(u)#:

#int 3^x dx = int e^ln(3^x) dx#

Use the property of logarithms that allows the exponent within the argument to become a factor on the outside:

#int 3^x dx = int e^(ln(3)x) dx#

Use the property #int e^(alphax)dx = 1/alphae^(alphax)+C#

#int 3^x dx = 1/ln(3)e^(ln(3)x)+C#

Substitute #e^(ln(3)x)= 3^x#

#int 3^x dx = 1/ln(3)3^x+C#