Concept preview

What you will be able to do

1. 01Apply the reverse-power rule $\displaystyle\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$ for every $n\neq -1$, and handle the single excluded case with $\displaystyle\int\frac1x\,dx=\ln|x|+C$.
2. 02Recall and use the basic antiderivatives of $\sin x$, $\cos x$, and $e^x$, together with the constant-multiple rule $\int k f = k\int f$ and the sum rule $\int (f+g)=\int f+\int g$, to antidifferentiate polynomials and combinations.
3. 03Read and write indefinite-integral notation $\int f(x)\,dx=F(x)+C$ correctly, always including the constant of integration $+C$, and explain what the constant represents.

To differentiate a power you multiply by the exponent and drop it by one: $\tfrac{d}{dx}x^{k}=k\,x^{k-1}$. Antidifferentiation reverses both moves — you RAISE the exponent by one and DIVIDE by the new exponent. That gives the reverse-power rule: $\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\qquad n\neq -1.$ You can verify it instantly by differentiating the right-hand side: $\tfrac{d}{dx}\left(\tfrac{x^{n+1}}{n+1}\right)=\tfrac{(n+1)x^{n}}{n+1}=x^{n}$, exactly the integrand. This single formula covers positive powers, negative powers, and fractional (root) powers alike. For example $\int x^5\,dx=\tfrac{x^6}{6}+C$; $\int x^{-2}\,dx=\tfrac{x^{-1}}{-1}+C=-\tfrac1x+C$; and $\int \sqrt{x}\,dx=\int x^{1/2}\,dx=\tfrac{x^{3/2}}{3/2}+C=\tfrac23 x^{3/2}+C$.