Math 22 Integration by Substitution and the General Power Rule

The General Power Rule

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

Guidelines for Integration by Substitution

1.Let $u$ be a function of $x$

2.Rewrite the integral in terms of the variable $u$

3.Find the resulting integral in terms of $u$

4.Rewrite the antiderivative as a function of $x$

5.Check your answer by differentiating (optional)

Exercises 1 Find the indefinite integral

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

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

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

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

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

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

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