# Further integrations

## Further integrations

**Fundamentals of Further Integration**

**Integration**is the process of finding the integral of a function, which often represents an area under a curve or a total quantity from a rate of change.- The
**fundamental theorem of calculus**states that differentiation and integration are opposite processes. - An
**indefinite integral**is a family of functions F that differentiate to give the original function. Written as ∫f(x) dx = F(x) + c. - A
**definite integral**is the net area under a curve between two points`a`

and`b`

. Written as ∫ (from a to b) of f(x) dx. - The
**power rule for integration**is the reverse of the power rule for differentiation, ∫x^n dx = (1/(n+1))x^(n+1) + c, for n ≠ -1.

**Methods of Integration**

**Substitution**is used when the integral contains a function and its derivative. The substitution u = g(x) simplifies the integral to the form ∫f(u) du.**Integration by parts**, based on the product rule for differentiation, is used for integrals of the form ∫u v dx. It’s often used when a product of functions is involved with one part being easily integrable and the other having a simpler derivative.**Partial fractions**are used when integrating rational functions (where the numerator has a lower degree than the denominator). By expressing the fraction as simpler fractions, more straightforward integrals are created.

**Integration of Trigonometric, Exponential, and Logarithmic Functions**

- Trigonometric functions can be integrated using Pythagorean identities and trigonometric substitution.
- The integrals of
**sin(x)**,**cos(x)**,**sec^2(x)**, and**csc^2(x)**are related to readily familiar functions. - The integral of
**e^x**is itself, while for a^x (a > 0, a ≠ 1), the integral is (a^x)/log(a). -
For **logarithmic functions**, integral of lnx is xln x - x.

**Applying Integration in Practical Contexts**

- Integration can be used for
**finding areas between curves**, by taking the difference between the integrals of two functions. - It is also used in
**solving differential equations**, particularly in physics, engineering and economics. - The
**mean value theorem for integrals**and the**first and second fundamental theorems of calculus**are also significant areas of application. - One common practical use is in calculating
**work done**, with the integral of force over distance. - It can also be used for calculating volume using
**disk method**or**shell method**.

**Numerical Integration**

- Sometimes it’s impossible or too difficult to find the exact area under a curve. In these instances, numerical integration, including the
**trapezium rule**,**midpoint rule**or**Simpson’s rule**, can provide an approximation.