Integral

In calculus, integrals measure the area under or over a curve, whether or not the boundaries at which the integral is taken are definite.

If the integral is indefinite (i.e. there are no boundaries), the integral works as follows:
 * $$\int n' \text{dx} = n + C$$

where C is the constant of integration.

If the integral is definite, the integral is formulated as follows:
 * $$\int_{a}^{b} F'(x) \text{dx} = \lim_{\delta \rightarrow \epsilon} f(x_{\delta}) \Delta x$$

where δ is the interval at which x increases and ε is an infinitesimal.

This can also be expressed as follows:
 * $$\int_{a}^{b} F'(x) \text{dx} = A_{f(x)} \vert_{a}^{b} = A_{f(b)} - A_{f(a)}$$

This is essentially the change in area under or over the curve from f(a) to f(b).

The derivative is the inverse of the integral, as shown below:
 * $$\frac{d}{dx} n = n'$$

This is completely generalized in the fundamental theorem of calculus.
 * $$\frac{d}{dx} \int_{a}^{b} f(t) \text{dx} = f(x)$$


 * $$\int_{a}^{b} F'(x) \text{dx} = F(x)\vert_{a}^{b} = F(b) - F(a)$$

Basic Properties

 * $$\int x^{n} \text{dx} = \frac{x^{n+1}}{n+1} + C$$


 * $$\int (\sqrt[n]{x})^{m} \text{dx} = \int x^{\frac{m}{n}} \text{dx} = \frac{x^{\frac{m}{n}+1}}{\frac{m}{n}+1} + C = \frac{x^{\frac{m+n}{n}}}{\frac{m+n}{n}} + C$$


 * $$\int c \text{dx} = cx + C$$ for any constant c.


 * $$\int a^{x} \text{dx} = \frac{a^{x}}{x \ln{a}} + C$$


 * $$\int \frac{1}{x} \text{dx} = \ln{|x|} + C$$


 * $$\int e^{x} \text{dx} = e^{x} + C$$


 * $$\int k*f(x) \text{dx} = k \int f(x) \text{dx}$$


 * $$\int [f(x) \pm g(x)] \text{dx} = \int f(x) \text{dx} \pm \int g(x) \text{dx}$$

Integration of Trig Functions

 * $$\int \sin{x} \text{dx} = -\cos{x} + C$$


 * $$\int \cos{x} \text{dx} = \sin{x} + C$$


 * $$\int \tan{x} \text{dx} = -\ln{|\cos{x}|} + C$$


 * $$\int \cot{x} \text{dx} = \ln{|\sin{x}|} + C$$

Scalar Integration and Integration of Sums/Differences
As seen above, multiplying an indefinite integral by a scalar works as follows:
 * $$\int k*f(x) \text{dx} = k \int f(x) \text{dx}$$

Here's an example of this property in action.
 * $$\int 2x^{2} \text{dx} = 2 \int x^{2} \text{dx}$$
 * $$= \frac{2x^{3}}{3} + C$$

Let's verify this result with a derivative, since
 * $$\frac{d}{dx} (\frac{u}{v}) = \frac{u'v - v'u}{v^{2}}$$

Let's plug our result into this formula.
 * $$\frac{d}{dx} (\frac{2x^{3}}{3}) = \frac{(6x^{2} * 3) - 0}{9}$$


 * $$= \frac{18x^{2}}{9} = 2x^{2}$$

As seen above, integration of the sum or difference of two functions works as follows:
 * $$\int [f(x) \pm g(x)] \text{dx} = \int f(x) \text{dx} \pm \int g(x) \text{dx}$$

Here's an example of this property in action.
 * $$\int (x^{3} + 2x) \text{dx} = \int x^{3} \text{dx} + \int 2x \text{dx} = \int x^{3} \text{dx} + 2 \int x \text{dx}$$
 * $$= \frac{x^{4}}{4} + \frac{2x^{2}}{2} + C = \frac{x^{4}}{4} + x^{2} + C$$

Let's verify this result with a derivative.
 * $$\frac{d}{dx} (\frac{x^{4}}{4} + x^{2}) = \frac{d}{dx} (\frac{x^{4}}{4}) + \frac{d}{dx} x^{2}$$


 * $$= \frac{(4x^{3} * 4) - 0}{16} + 2x = \frac{16x^{3}}{16} + 2x = x^{3} + 2x$$

From this, you can see that the property of sums/differences applies to derivatives as well as integrals.