Welcome to our tutorial on integration in A-level CIE Pure Mathematics 1. Integration is the reverse process of differentiation. It allows us to find the area under a curve and solve differential equations. The basic integration formula states that the integral of x to the power of n equals x to the power of n plus 1, divided by n plus 1, plus a constant C. This formula works for all values of n except negative 1. Let's look at an example: the integral of x squared from 1 to 3. We apply the formula to get x cubed over 3, evaluated between the limits 1 and 3. This gives us 27 over 3 minus 1 over 3, which equals 26 over 3.
In A-level CIE Pure Mathematics 1, you'll encounter several integration techniques. The first is direct integration, where we apply the standard formula for powers of x. The second technique is integration by substitution, which is useful when we can simplify a complex integral by making a substitution. Let's look at an example: the integral of 2x times (x squared plus 1) cubed. We can substitute u equals x squared plus 1, which means du equals 2x dx. This transforms our integral into the integral of u cubed du, which equals u to the power of 4 divided by 4 plus C. Substituting back, we get (x squared plus 1) to the power of 4 divided by 4 plus C. The third technique is integration by parts, which is used when integrating products of functions.
Let's explore integration by parts in more detail. The formula states that the integral of u dv equals u times v minus the integral of v du. This technique is particularly useful when integrating products of functions. The key is to choose u and dv strategically to simplify the resulting integral. Let's work through an example: the integral of x times cosine of x. We choose u equals x and dv equals cosine of x dx. This means du equals dx and v equals sine of x. Applying the formula, we get x times sine of x minus the integral of sine of x dx. The integral of sine of x is negative cosine of x, so our final answer is x times sine of x plus cosine of x plus C. On the graph, we can see the original function x cosine of x in blue, and its integral x sine of x plus cosine of x plus C in red.
Now let's discuss definite integrals and their applications. A definite integral is evaluated between specific limits, a and b, and is calculated as F of b minus F of a, where F is the antiderivative of f. Definite integrals have numerous applications in mathematics and physics. One common application is finding the area under a curve. We can also use definite integrals to calculate the area between two curves. In our example, we're finding the area between f of x equals x squared and g of x equals x, from x equals 1 to x equals 3. To do this, we integrate the difference of the functions: the integral from 1 to 3 of x squared minus x dx. Evaluating this, we get x cubed over 3 minus x squared over 2, evaluated between 1 and 3. This gives us approximately 4.67 square units. Other applications include calculating volumes of revolution and determining work and energy in physics problems.
Let's conclude with some key points and exam tips for A-level CIE Pure Mathematics 1 integration. Remember that integration is the reverse process of differentiation. The three key techniques you need to master are direct integration, integration by substitution, and integration by parts. For definite integrals, always evaluate F of b minus F of a. And don't forget to include the constant of integration C for indefinite integrals. When preparing for exams, here are five important tips: First, check your integration answer by differentiating it. Second, draw diagrams for area problems to visualize what you're calculating. Third, choose u and dv strategically when using integration by parts. Fourth, watch for special cases, such as the integral of 1 over x, which gives ln of the absolute value of x. And finally, always include the constant of integration C for indefinite integrals. Practice with a variety of functions and applications to build your confidence and skills.