-
How do you antidifferentiate a function? Antidifferentiation (also called integration) is the opposite operation to differentiation. If you have the differentiation of a function, you can then obtain the original function via integration (antidifferentiation). Given a derivative \(f^{\prime}\left(x\right)\) of a function we want to find the original function \(f\left(x\right)\). The...
-
How do you integrate a polynomial where x is raised to a power? We saw this in the previous section on antidifferentiation. But how do you integrate a linear expression in brackets where the whole bracket is raised to a power? This module shows how to integrate functions like :...
-
How do you integrate a logarithm? How do you integrate an exponential function? How do you integrate a trigonometric function? Read this worksheet to see how this is done. This module looks at integrals such as \(\int\frac{1}{x}dx\) and\(\int\frac{1}{3x-1}dx.\) The power rule for integration One of the most important rules for...
-
The integration of exponential functions is widely used in various fields such as physics, engineering, and economics to model and solve problems involving growth, decay, and other processes that change exponentially over time. Keep reading to find out more. This module deals with differentiation of exponential functions such as: \[\begin{align*}...
-
The integration of trigonometric functions in solving problems is related to oscillatory motion, waves and other periodic phenomena in physics and engineering. This module deals with integration of trigonometric functions. These include: \[\begin{align*} & \int\sin\left(2x+3\right)dx\\ & \int\cos\left(5x\right)dx\\ & \int_{1}^{2}\sec^{2}\left(x-2\right)dx. \end{align*}\] Indefinite integral (antiderivative) of a trigonometric function Recall that: \[\begin{align*}...
-
An area under a curve might be above the axis (and therefore positive). But sections might also be below the axis (and therefore negative). Read this worksheet to see how to deal with finding the integral for sections of graph which go above and below the x-axis (horizontal axis). This...
-
An expression that is composed of two functions (say an algebraic expression nested within a trigonometric expression) can be complicated to integrate. You can simplify this by substituting a single pronumeral (say u) to represent one of the functions. The substitution rule for integration is like the chain rule for...
-
How do you integrate an expression when there is an algebraic expression in the numerator and denominator of a fraction? Integrating using partial fractions helps you to solve this problem. Read this worksheet for several worked examples. Some times a complex function may be integrated by breaking it up into...
-
If you can consider your expression to be a product (i.e. multiplication \(\times\)) of two functions, you can integrate this using Integration by parts. This reflects the product rule in differentiation and is applicable to logarithmic, exponential, trigonometric and algebraic functions. Integration by parts is a technique for integrating the...
-
Integrating will find the area between the curve and the x-axis (horizontal axis). We've learned how to limit this to a section of the x axis. But what if you have an area that is bounded by limits on both the x and the y axes? You will need to...
-
Integration is vital in engineering. It is the key mathematical tool for finding the centre of mass or the surface area of a body. Integration is also called antidifferentiation. It is the reverse process of differentiation. If you differentiate an expression, you can integrate it to get back to the...
