Integration

• Integration as the reverse of differentiation

If ^dy/_dx = xⁿ, then

y=\frac{1}{n+1} x^{n+1}+c, n \neq-1

• Indefinite Integration

\int k f(x) d x=h \int f(x) dx

\int(f(x)+g(x)) d x=\int f(x) d x+\int g(x) d x

• Definite Integration of the form xⁿ

\int_{a}^{b} x^{n} d x=\left[\frac{x^{n+1}}{n+1}\right]_{a}^{n}=\frac{b^{n+1}-a^{n+1}}{n+1}, n \neq 1

• Area Under a Curve

The value of the definite integral represents the area enclosed within the x-axis, the function and the two limits.

When a graph goes below the x-axis, the corresponding values of y are negative, so the area becomes negative.

To find the area where some parts of the curve are above the axis and others are below it, you need to separate the integrals so that the negative and positive values do not cancel each other out.