Example 1: Indefinite Integral Inverse Trig

$\int \frac{1}{1+x^2}\mathit{ dx}$

Step 1: Simplify and look for algebraic rewrites.

None in this example.

$\int \frac{1}{1+x^2}\mathit{ dx}$

Step 2: Identify any term(s) that include one of the six inverse trig function special cases.

Here you have 1 chunk, and it is one of the inverse trig function special cases.

$\int \frac{1}{1+x^2}\mathit{ dx}$

Step 3: Take the antiderivativeof the inversetrig function special cases using their specific Recipe.

Chunk 1: $\frac{1}{1+x^2}$

$\int \frac{1}{1+x^2}\mathit{ dx}={\mathit{tan}}^{–1}\left(x\right)\mathit{+}\mathit{C}$

DO NOT FORGET THE +C

Final Result Meaning: Remember the Definite Integral will always provide you a definite value , and the Indefinite Integral provides you a family of solutions .

The antiderivative of the equation $f\prime \left(x\right)=\frac{1}{1+x^2}$is the family of graphs $f\left(x\right)={\mathit{tan}}^{–1}\left(x\right)\mathit{+}\mathit{C}$.

All graphs of the form $f\left(x\right)={\mathit{tan}}^{–1}\left(x\right)\mathit{+}\mathit{C}$ have the same derivative (instantaneous rate of change), $f\prime \left(x\right)=\frac{1}{1+x^2}$, at any x-value .

The only difference between any of original graphs, $f\left(x\right)={\mathit{tan}}^{–1}\left(x\right)\mathit{+}\mathit{C}$, is just a vertical shift of +C .

In the graph above are examples of 3-different C-values. You can see the instantaneous rate of change ( slope ) is the same at every x-value between all the different examples, no matter the +C .