Quick Question on $\int_{0}^{c}(\frac{x}{c})^{n}dx$ Apparently $\int_{0}^{c}(\frac{x}{c})^{n}dx=\frac{c}{n+1}$ but I do not see why: 
I mean $\int_{0}^{c}(\frac{x}{c})^{n}dx=\frac{1}{n+1}(\frac{x}{c})^{n+1}\vert_{0}^{c}=\frac{1}{n+1}(\frac{c}{c})^{n+1}=\frac{1}{n+1}$
 A: As the comment suggest, you've forgotten to take out the coefficient before integrating.
$$\int_{0}^{c}(\frac{x}{c})^{n}dx=\frac1{c^n}\int_{0}^{c}x^{n}dx=\frac1{c^n}\frac{x^{n+1}}{n+1}\Big\vert_{0}^{c}=\frac1{c^n}\frac{c^{n+1}}{n+1}=\frac{c}{n+1}$$
A: The derivative of ${1\over n+1}({x\over c})^{n+1}$ is ${1\over c^{n+1}}{{n+1}\over {n+1}}x^n$
A: Well, there is one thing in your solution that is wrong. When you integrate x/c, you must divide the resulting integral by the derivative of the integrand (the integrand is the expression under the integral which, in this case, is x/c) with respect to the variable of integration. In this case, the derivative of x/c with respect to x is 1/c. As 1/(1/c) = c, hence you must multiply your answer by c.
Alternatively, you could take out the constant of (1/c)^n out of the integral and then integrate the function.
Hope this helped! And sorry for the non-mathematical representation, as I am still to learn how to type mathematical expression in SE.
A: Let $y=x/c$ so your integral is $\int_{y=0}^{y=1}y^n dx$. What you've evaluated is $\int_{y=0}^{y=1}y^n dy$. But $dx=d(cy)=cdy$, so there's an extra factor of $c$.
