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

Question

$\int x^n\,\mathrm dx=\dfrac{x^{n+1}}{n+1}+C\qquad n\ne-1$ $\int(ax+b)^n\,\mathrm dx=\dfrac{(ax+b)^{n+1}}{a(n+1)}+C\qquad a\ne0,\ n\ne-1$ The second formula requires a linear inner function. It does not apply directly to $\int(x^2+1)^5\,\mathrm{d}x$. Example 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.

Accepted Answer

$\displaystyle\int x^n\,\mathrm{d}x=\dfrac{x^{n+1}}{n+1}+C$ for $n\ne-1$, and $\displaystyle\int(ax+b)^n\,\mathrm{d}x=\dfrac{(ax+b)^{n+1}}{a(n+1)}+C$ for $a\ne0$ and $n\ne-1$

Integration of Power Functions

Integration of Power Functions

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