xn, dx= xn+1n+1+C n!= -1 (ax+b)n, dx= (ax+b)n+1a(n+1)+C a 0, n!= -1 The second formula requires a linear inner function. It does not apply directly to (x2+1)5,dx. Example Integrate 6x2-1 with respect to x. (6x2-1),dx =6(x33)-x+C =2x3-x+C Integrate (2x+1)5 with respect to x. (2x+1)5,dx = (2x+1)62(6)+C = (2x+1)612+C Integrate x4-2x3+1 with respect to x. (x4-2x3+1),dx = x55-2(x44)+x+C = x55- x42+x+C Use intervals where the integrand is real and defined. If a = 0, integrate the resulting constant. Check by differentiating and include the integration constant.

\(\int x^n\,\mathrm dx=\dfrac{x^{n+1}}{n+1}+C\qquad n\ne-1\)

\(\int(ax+b)^n\,\mathrm dx=\dfrac{(ax+b)^{n+1}}{a(n+1)}+C\qquad a\ne0,\ n\ne-1\)

The second formula requires a linear inner function. It does not apply directly to \(\int(x^2+1)^5\,\mathrm{d}x\).

Integrate \(6x^2-1\) with respect to \(x\).

\(\int(6x^2-1)\,\mathrm{d}x\)\({}=6\left(\dfrac{x^3}{3}\right)-x+C\)\({}=2x^3-x+C\)

Integrate \((2x+1)^5\) with respect to \(x\).

\(\int(2x+1)^5\,\mathrm{d}x\)\({}=\dfrac{(2x+1)^6}{2(6)}+C\)\({}=\dfrac{(2x+1)^6}{12}+C\)

Integrate \(x^4-2x^3+1\) with respect to \(x\).

\(\int(x^4-2x^3+1)\,\mathrm{d}x\)\({}=\dfrac{x^5}{5}-2\left(\dfrac{x^4}{4}\right)+x+C\)\({}=\dfrac{x^5}{5}-\dfrac{x^4}{2}+x+C\)

Use intervals where the integrand is real and defined. If a = 0, integrate the resulting constant. Check by differentiating and include the integration constant.

Integration of Power Functions

Integration of Power Functions

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