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# Integration of Algebraic Functions

## Definition

The integral of the expression $$x^n$$ where $$n \neq -1$$ is:

$\int x^ndx = \frac{x^{n+1}}{n+1} + C$

The integral of $$x^{-1}$$ is:

$\int \frac{1}{x}\,dx = \ln \left| x \right| + C$

Note the absolute value inside the natural log.

The integral of the exponential function $$e^x$$ is:

$\int e^x \, dx = e^x + C$

## Technique

### Find a: $$\displaystyle \int_0^{\ln a} e^x \, dx = 100$$

Since $$\int e^x \, dx = e^x + C$$, we see that:

\begin{align} \int_0^{\ln a} e^x \, dx &= 100 \\ \left[ e^x \right]_0^{\ln a} &= 100 \\ e^{\ln a}-e^0 &= 100 \\ a &= 101 _\square \end{align}

### Evaluate: $$\displaystyle \int_0^1 7x^6 + 42 + \frac{2}{x^3} \, dx$$.

We can rewrite this problem so that all terms are in the form $$x^n$$:

$\displaystyle \int_0^1 7x^6 + 42x^0 + 2x^{-3} \, dx$

Now, let's integrate, using the rules above:

\begin{align} \int_0^1 7x^6 + 42 + 2x^{-3} \, dx &= \int_0^1 7x^6 dx + \int_0^1 42 x^0 \, dx + \int_0^1 2x^{-3}\, dx \\ &= \left[ 7\left(\frac{x^{6+1}}{6+1} \right ) + 42x^1 + 2 \left( \frac{x^{-3+1}}{-3+1} \right ) \right]_0^1 \\ &=\left. x^7 + 42x -x^{-2} \right|_0^1 \\ &= \left(1^7 +42(1)-1\right)-\left(0\right) \\ &= 42 \, _\square \end{align}

Note by Arron Kau
2 years, 7 months ago