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# c4 integration help 3^2x

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1. how do i integrate 3^2x
2. (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.
3. you can write 3 as eln3....
4. Correct me if I'm wrong, but i think this is how you do it (see attached). Obviously '+ C'
Attached Images

5. (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 :-)
6. (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 :-)
7. (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 :-)

Also, moved to maths.
8. (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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