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# Calculus

## Indefinite Integral

As noted in the previous section, the infinite many antiderivatives (with different constants) of a given function $$f(x)$$ is called an indefinite integral and is written $$\int f(x) dx$$.

The indefinite integral of a function $$f(x)$$ can be defined as the antiderivative $$F(x)$$ plus any constant $$c$$,

$\int f(x)dx = F(x) + c$

## Theorems of Linearity

1. If $$f$$ and $$g$$ are are continuous functions, then

$\int \left(f(x) \pm g(x)\right) dx = \int f(x) dx \pm \int g(x) dx$

2. If $$f$$ is a continuous function and $$k$$ is a real constant, then

$\int kf(x) dx = k\int f(x) dx$

## Power Rule for Integration

To find the differential of a power of a function, we multiply by the power, subtract 1 from it, and multiply by the differential of the function.

$\qquad \frac{d}{dx} x^n = \frac{x^{n-1}}{n-1}$

To find the integral, we reverse this procedure to get the power rule for integration:

$\qquad \int u^n du = \frac{u^{n+1}}{n+1} + c \qquad (n \neq -1)$

## Examples

Find the following integrals:

$\int 3A\sqrt{A}-5A^2)dA$

Solution:

Simplify square roots into fractional exponents and apply addition rules for integrals.

$=\int 3A^{\frac{3}{2}}dA - \int 5A^2dA$

Move constants outside integrals

$=3\int A^{\frac{3}{2}}dA - 5 \int A^2dA$

Integrate using power rule for integration and don't forget the CONSTANT

$=3\frac{A^{\frac{5}{2}}}{{\frac{5}{2}}} - 5{\frac{A^3}{3}} + c$

Simplify

$= \frac{6A^{\frac{5}{2}}}{5} - \frac{5A^3}{3} + c$

For more complex powers you can use a substitution before applying the power rule.

$\int (x^2 -x)\left(x^3 - \frac{3}{2}x^2)^8\right) dx$

Solution:

Substitute the values inside the bracket and find the derivative of the substitution

$Let \, u = x^3 - \frac{3}{2}x^2, du = (3x^2 - 3x)dx$

Substitute u and du in for x and dx in original integral

$\int \left(\underbrace{x^3 - \frac{3}{2}x^2}_\text{u})^8\right) \underbrace{(x^2 -x)dx}_\text{du}$

$= \int u^8du$

$= \frac{u^9}{9} + c$

Substitute x values back

$=\frac{\left(x^3 - \frac{3}{2}x^2 \right)^9}{9} + c$