I think I see where you are coming from now. The problem is the 13x in the denominator.
Your Short cut way:
int[ (2x+3)^12]dx
= (1/13) (2x+3)^13 / 2 ---> divide by the inner derivative of 2x+3 which is 2 to undo
the chain rule. Not x as you were doing
= (1/26) (2x+3)^13 + c
Check by taking the derivative:
d/dx [ (1/26)(2x+3)^13]
= (1/26) (13) [(2x+3)^12)] (2) --------> times by 2 as the result of the chain rule.
= (26/26)(2x+3)^12
= (2x+3)^12
I should point out that this little short cut only works when the inner derivative is a constant - in this case 2. If you are integrating ( 2x^2 + 3x)^13 then you are SOL.
@_ sunflower Yes, if the inner function is linear.
But your short cut can be used in other special cases. For example if you wanted to integrate: 2x^2*(x^3+5)^1/2. You will notice that the inner derivative is 3x^2 and you conveniently have a factor of 2x^2 out in front so just using the power rule on
(x^3 + 5)^ (1/2) to give (x^3+5)^(3/2) your outside factor be out by a constant when you use the chain rule. So take the derivative of your first attempt to see what that constant is. Don't divide by 3x^2.
So you can see that you have a factor 9/2 out in front instead of the desired 2 in the original integral so to change (9/2) into 2 just multiple by (4/9)
Finally: integral of 2x^2 *(x^3+5)^1/2 = (4/9)*(x^3+5)^3/2 + c
Remember, differentiating your final answer is the ultimate test.
I think I see where you are coming from now. The problem is the 13x in the denominator.
Your Short cut way:
int[ (2x+3)^12]dx
= (1/13) (2x+3)^13 / 2 ---> divide by the inner derivative of 2x+3 which is 2 to undo the chain rule. Not x as you were doing
= (1/26) (2x+3)^13 + c
Check by taking the derivative:
d/dx [ (1/26)(2x+3)^13]
= (1/26) (13) [(2x+3)^12)] (2) --------> times by 2 as the result of the chain rule.
= (26/26)(2x+3)^12
= (2x+3)^12
I should point out that this little short cut only works when the inner derivative is a constant - in this case 2. If you are integrating ( 2x^2 + 3x)^13 then you are SOL.
Thank you for your kind and amazing reply again.
Yeah I made a careless mistake there hahahahaha.
So the same idea goes to the integration of tan^2x? The inner derivative must be constant when using the short cut?
@_ sunflower Yes, if the inner function is linear.
But your short cut can be used in other special cases. For example if you wanted to integrate: 2x^2*(x^3+5)^1/2. You will notice that the inner derivative is 3x^2 and you conveniently have a factor of 2x^2 out in front so just using the power rule on
(x^3 + 5)^ (1/2) to give (x^3+5)^(3/2) your outside factor be out by a constant when you use the chain rule. So take the derivative of your first attempt to see what that constant is. Don't divide by 3x^2.
d/dx[ (x^3 + 5)^3/2] = (3/2)*(x^3 + 5)^1/2 * 3x^2 = (9/2) x^2*( x^3 + 5)^1/2
So you can see that you have a factor 9/2 out in front instead of the desired 2 in the original integral so to change (9/2) into 2 just multiple by (4/9)
Finally: integral of 2x^2 *(x^3+5)^1/2 = (4/9)*(x^3+5)^3/2 + c
Remember, differentiating your final answer is the ultimate test.
d/dx[ (4/9) *(x^3+5)^3/2] = (4/9)*(3/2)*(x^3+5)^1/2 * (3x^2) = 2x^2(x^+5)^(1/2)
This short cut only works if the outside factor is a constant*inner derivative.
@Ian Fowler Thank you sooo sooo much for you kindness!! Now I understand and know that I should be careful with the constant!
Since tan(x) whole square is not equal to {tan(tanx)}^2 thats why the second one is absolutely wrong.
Since tan(x) whole square is not equal to {tan(tanx)}^2 thats why the second one is absolutely wrong.