This lecture from M.I.T. is 48 minutes long and covers methods of differentiation and integration using trigonometric functions. It is aimed at undergraduate students.
These lectures will give you some idea of how this subject is treated at university level.
Summary/Background
You will not succeed. Though the derivative of e^{x^2} is 2xe^{x^2} , you cannot integrate e^{x^2} by basic methods. Neither can you integrate e^{-x^2} . Note than in Statistics module S1, the equation of the normal distribution involves the latter function. To find normal probabilities you have to find areas under the curve. For this you use normal probability tables - not integration.
So, how is e^{x^2} integrated? Go to a site like The Integrator, enter the function and you will be told the integral is \frac{1}{2} \sqrt{\pi} Erfi(x) where Erfi(x) is the "imaginary error function". Find out more at University!
Software/Applets used on this page
Free lecture notes, exams, and videos are available from Massachusetts Institute of Technology at MIT. No registration required.Glossary
derivative
rate of change, dy/dx, f'(x), 
, Dx.
, Dx.differentiation
The process of finding a derivative.
equation
A statement that two mathematical expressions are equal.
function
A rule that connects one value in one set with one and only one value in another set.
integral
the anti-derivative
integration
the process of finding an integral, the reverse process to differentiation.
normal
A line passing through a curve at a point that is perpendicular to the tangent to the curve at that same point.
normal distribution
A key distribution in statistics which is used to model many naturally occurring phenomena, such as heights and weights; its distribution is bell-shaped.
This question appears in the following syllabi:
| Syllabus | Module | Section | Topic | Exam Year |
|---|---|---|---|---|
| AP Calculus AB (USA) | 4 | Integration | XTRA | - |
| AP Calculus BC (USA) | 4 | Integration | XTRA | - |
| AQA A-Level (UK - Pre-2017) | C3 | Integration | XTRA | - |
| CCEA A-Level (NI) | C4 | Integration | XTRA | - |
| CIE A-Level (UK) | P3 | Integration | XTRA | - |
| Edexcel A-Level (UK - Pre-2017) | C4 | Integration | XTRA | - |
| I.B. Higher Level | 6 | Integration | XTRA | - |
| I.B. Standard Level | 6 | Integration | XTRA | - |
| Methods (UK) | M9 | Integration | XTRA | - |
| OCR A-Level (UK - Pre-2017) | C4 | Integration | XTRA | - |
| OCR-MEI A-Level (UK - Pre-2017) | C3 | Integration | XTRA | - |
| Pre-U A-Level (UK) | 5 | Integration | XTRA | - |
| Universal (all site questions) | I | Integration | XTRA | - |
| WJEC A-Level (Wales) | C4 | Integration | XTRA | - |