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c4 integration help 3^2x

how do i integrate 3^2x
Original post by imran_
how do i integrate 3^2x

you know that the derivative of a^x is a^xlna. Therefore, the derivative of 3^x is 3^xln3. In the case of 3^2x it is 3^2xln3(2), it is multiplied by 2 because you have to do the chain rule as 3 is to a power of x but in this case the coefficient of x isn't 1, but rather 2 (and the differential of 2x is 2 so you have to take that into account). Knowing this you can see that 3^2xln3(2) is similar to 3^2x (what you are trying to integrate, but not the same. You have two constants there (ln3) and 2 which you don't want. You will want to get rid of them by going by to the 3^2x line and putting a 1/ln3 * 1/2 there so that when you differentiate (1/2ln3)*3^2x the 2 will cancel with the 1/2 and the 1/ln3 will cancel with the ln3. If that doesn't make sense I can maybe explain that differently.
(edited 7 years ago)
you can write 3 as eln3....
Correct me if I'm wrong, but i think this is how you do it (see attached). Obviously '+ C'
(edited 7 years ago)
Reply 4
Original post by alexcapstick97
Correct me if I'm wrong, but i think this is how you do it (see attached). Obviously '+ C'


can you explain how you integrated please, I understand how you got e^2xln3

thanks by the way :-)
Reply 5
Original post by OrionMusicNet
you know that the derivative of a^x is a^xlna. Therefore, the derivative of 3^x is 3^xln3. In the case of 3^2x it is 3^2xln3(2), it is multiplied by 2 because you have to do the chain rule as 3 is to a power of x but in this case the coefficient of x isn't 1, but rather 2 (and the differential of 2x is 2 so you have to take that into account). Knowing this you can see that 3^2xln3(2) is similar to 3^2x (what you are trying to integrate, but not the same. You have two constants there (ln3) and 2 which you don't want. You will want to get rid of them by going by to the 3^2x line and putting a 1/ln3 * 1/2 there so that when you differentiate (1/2ln3)*3^2x the 2 will cancel with the 1/2 and the 1/ln3 will cancel with the ln3. If that doesn't make sense I can maybe explain that differently.


the c3 formula states that a^x is e^xlna
so in my case it'll be e^2xln3

after this how would I integrate this?

thanks :-)
Reply 6
Original post by imran_
the c3 formula states that a^x is e^xlna
so in my case it'll be e^2xln3

after this how would I integrate this?

thanks :-)


eaxdx=eaxa+C\int e^{ax} \, \mathrm{d}x = \frac{e^{ax}}{a} + \mathcal{C}

Also, moved to maths.
Reply 7
Original post by OrionMusicNet
you know that the derivative of a^x is a^xlna. Therefore, the derivative of 3^x is 3^xln3. In the case of 3^2x it is 3^2xln3(2), it is multiplied by 2 because you have to do the chain rule as 3 is to a power of x but in this case the coefficient of x isn't 1, but rather 2 (and the differential of 2x is 2 so you have to take that into account). Knowing this you can see that 3^2xln3(2) is similar to 3^2x (what you are trying to integrate, but not the same. You have two constants there (ln3) and 2 which you don't want. You will want to get rid of them by going by to the 3^2x line and putting a 1/ln3 * 1/2 there so that when you differentiate (1/2ln3)*3^2x the 2 will cancel with the 1/2 and the 1/ln3 will cancel with the ln3. If that doesn't make sense I can maybe explain that differently.


Nevermind, I got it. Thanks a lot!

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