Question

Integrate 7^x * 5^x with respect to x without using the property implying a^x * b^x = (ab)^x. @Calculus1

Answer 1

\[\int\limits{35^x}dx = \frac{35^x}{\ln35}+c\]

Answer 2

For a > 0,
\[ \frac{d \ }{dx} a^x = \ln a . a^x \]
Thus integrating by parts
\[ J = \int 7^x 5^x \ dx = \frac{1}{\ln 7} 7^x 5^x - \frac{\ln 5}{\ln 7} \int 7^x 5^x \ dx \]
This last integral is a constant times the original integral J.

Now solve algebraically for J.

Answer 3

@sarah, yes exactly. But the question explicitly asks that we don't reduce \(7^x5^x\) to \(35^x\).

Answer 4

@James, is this an instance of when the integral repeats?

Answer 5

If you will, yes.

Answer 6

It's a slightly tortured question and Sarah's approach is the natural one. The good news is you can check your answer against hers.

Answer 7

I don't know if my Algebra is extremely terrible or I've misunderstood your directions, but when I attempted to solve the problem my answer resulted in Ln 7 + Ln 5 = 7^x * 5^x...

Answer 8

What you should find is that
\[ (\ln 7 + \ln 5) J = 7^x5^x \]
i.e.,
\[ J = \frac{1}{\ln 7 + \ln 5} 7^x5^x \]

Answer 9

Now check and make sure this is the same as Sarah's (correct) answer.

Answer 10

Sarah's answer is equal to yours. What I don't comprehend is how you use the variable for your answer in your equation.

Answer 11

are you talking about his integrating by parts technique?

Answer 12

i'm not really sure what you are saying

Answer 13

\[ \int\limits 7^x 5^x \ dx = \frac{1}{\ln 7} 7^x 5^x - \frac{\ln 5}{\ln 7} \int\limits 7^x 5^x \ dx \]

now we are trying to evaluate \[\int\limits_{}^{}7^x 5^x dx\]

do you know how to solve
y=a-cy for y?

y+cy=a
y(1+c)=a
y=a/(1+c)


so just Let
\[J=\int\limits_{}^{}7^x5^x dx\]
\[ J = \frac{1}{\ln 7} 7^x 5^x - \frac{\ln 5}{\ln 7} J\]

solve this equation for J

Answer 14

so add \[\frac{\ln(5)}{\ln(7)} J\]
on both sides

Answer 15

I got it myininaya and James, thanks.

Answer 16

\[\frac{\ln(5)}{\ln(7)}J+J=\frac{1}{\ln(7)} 7^x5^x\]
oh ok....
nvm then

Answer 17

just don't forget to put pluc C somewhere

Answer 18

I definitely won't forgot that myininaya!

Answer 19

ok lol

Answer 20

I am sorry James i misread it :(