# Math 22 Integration by Substitution and the General Power Rule

## The General Power Rule

 If ${\displaystyle u}$ is a differentiable function of ${\displaystyle x}$, then
${\displaystyle \int u^{n}{\frac {du}{dx}}dx=\int u^{n}du={\frac {u^{n+1}}{n+1}}+C}$ for ${\displaystyle n\neq -1}$


## Guidelines for Integration by Substitution

1.Let ${\displaystyle u}$ be a function of ${\displaystyle x}$

2.Rewrite the integral in terms of the variable ${\displaystyle u}$

3.Find the resulting integral in terms of ${\displaystyle u}$

4.Rewrite the antiderivative as a function of ${\displaystyle x}$

Exercises 1 Find the indefinite integral

1) ${\displaystyle \int (5x^{2}+1)^{2}(10x)dx}$

Solution:
Let ${\displaystyle u=5x^{2}+1}$, so ${\displaystyle du=10xdx}$
${\displaystyle \int (5x^{2}+1)^{2}(10x)dx=\int u^{2}du={\frac {u^{2+1}}{2+1}}+C={\frac {u^{3}}{3}}+C={\frac {(5x^{2}+1)^{3}}{3}}+C}$

2) ${\displaystyle \int {\sqrt {1-x^{2}}}(-2x)dx}$

Solution:
Let ${\displaystyle u=1-x^{2}}$, so ${\displaystyle du=-2xdx}$
${\displaystyle \int {\sqrt {1-x^{2}}}(-2x)dx=\int {\sqrt {u}}du=\int u^{\frac {1}{2}}du={\frac {u^{{\frac {1}{2}}+1}}{{\frac {1}{2}}+1}}+C={\frac {u^{\frac {3}{2}}}{\frac {3}{2}}}+C={\frac {2}{3}}u^{\frac {3}{2}}+C={\frac {2}{3}}(1-x^{2})^{\frac {3}{2}}+C}$

3) Find an equation of the function ${\displaystyle f(x)}$ that has the given derivative ${\displaystyle f'(x)=2x(4x^{2}-10)^{2}}$ and whose graph passes through the point ${\displaystyle (2,10)}$

Solution:
Let ${\displaystyle u=4x^{2}-10}$, so ${\displaystyle du=8xdx}$, so ${\displaystyle dx={\frac {du}{8x}}}$
${\displaystyle f(x)=\int f'(x)dx=\int 2x(4x^{2}-10)^{2}dx=\int 2x(u)^{2}{\frac {du}{8x}}=\int {\frac {1}{4}}u^{2}du={\frac {1}{4}}\int u^{2}du={\frac {1}{4}}{\frac {u^{3}}{3}}+C={\frac {1}{12}}(4x^{2}-10)^{3}+C}$
Given ${\displaystyle f(x)}$ passes through ${\displaystyle (2,10)}$, so ${\displaystyle x=2,y=10}$ satisfy the equation ${\displaystyle f(x)={\frac {1}{12}}(4x^{2}-10)^{3}+C}$.
So, ${\displaystyle 10={\frac {1}{12}}(4(2)^{2}-10)^{3}+C}$, hence ${\displaystyle C=-8}$
Therefore, ${\displaystyle f(x)={\frac {1}{12}}(4x^{2}-10)^{3}-8}$.