# Introduction to integration

Watch this video for an introduction to integration.
4.1
In this video, we are going to look at some further integration. So we’re going to first define some rules for integrating trigonometric functions, which are that the integral of sine kx with respect to x is negative cosine kx over k plus some constant. And the integral of cosine of kx with respect to x is sine kx divided by k plus some constant C. So we have the following. The differential of a function y is that dy over dx is 8 sine of 4x. And we know that when x is 0, y is equal to 4. Can we find the function y?
48
So we’re first going to use the rule that y is the integral of the first derivative of y with respect to x, integrated with respect to x, so we’re going to integrate 8 sine of 4x with respect to x. And using that top rule, we have that k is 4. So it is negative 8 cosine 4x divided by 4 plus some constant C, which simplifies to negative 2 cosine of 4x plus C. So we’re going to use now, the fact that when x is 0, y is equal to 4 to find what C is, the constant. So we’ve got that y is the following.
91.3
So if we substitute in y equals 4 and x equals 0, we get 4 is negative 2 cosine 4 multiplied by 0 plus the constant C. And we know the cosine of 0 is 1, so 4 is equal to negative 2 plus C. So 4 is equal to negative 2 plus C, which means C must be equal to 6 in this case. And so our final answer is that y is negative 2 cosine of 4x plus the constant 6. So we’re now going to look at integrating exponential functions.
131.4
Well we know, the definition we have to know, is that if we integrate n e to the power of kx with respect to x, then that is equal to n e to the power of kx divided by k plus some constant C. So if we wanted to find the integral of 4e to the power of negative 3x with respect to x, what we have in this case, n is 4. In our definition, k is negative 3. So the integral, when we integrate, gets those 4e to the negative 3x divided by negative 3 plus C, using the definition, which is negative 4/3e to the power of negative 3x plus C.
177.7
In question three, we’re going to look at integrating 1 over x. So the definition that we must know is that if we integrate 1 over x with respect to x, then this is the same as integrating x to the power of negative 1, which is the natural logarithm of x plus C. So we’ll go it. We’ve got curve such that first derivative of y respect to x is x to the power of negative 1 plus 3. And we know the point 1, negative 1 lies on the curve. Can we find the equation for y? So we’re going to integrate x to the power of negative 1 plus 3 to get what y is.
217.7
And we can split that into two separate integrals. So when we integrate x to the negative 1 with respect to x, using the definition, that’s the natural logarithm of x. And we also know that if we integrate some constant with respect to x, then it is kx plus C. So if we integrate 3, k is equal to 3. So we have plus 3x plus C. And we’ve got the point x is 1, y is negative 1, lies on the curve. So we can find C by substituting the values in.
256.1
So we would have that negative 1 is the natural logarithm of 1, plus 3 multiplied by x is 1, plus C, which is the same as negative 1 is 0 plus 3 plus C. So negative 1 is 3 plus C. So C is negative 4. Such that we can write our final answer as y is the natural logarithm of x plus 3x plus C, which in this case is negative 4. So we’re finally going to look at the reverse chain rule. Now that is the definition of the chain rule from differentiation. And we know the integration is the opposite of differentiation, so we can just write that instead.
302.6
The integration of the final answer in the chain rule takes us back to what we differentiated, f of g of x plus some constant C because we’re integrating. So we’re going to integrate the following. Now this can be written in the form of the reverse chain rule. So we would have that f dash of x, the first derivative of f is sine of x, because we’ve got sine of x squared. Now in the integral, we need f dash of gx. So instead of sine x, we need the sine of x squared, which means gx must be x squared.
346.6
So the first derivative of g would be 2x, because if gx is x squared, if we differentiate that, that would be 2x. And we can also find f of x by integrating f of dash x. So we’re integrating that sine of x, which is negative cosine of x plus C. So using all of this information, we know that if we integrate something in this form, using the reverse chain rule, it’s f of x plus C. When we’ve got f, we’ve got g. And so, the integral would be f of x, which is negative cosine x plus C. But instead of x, it’s f of gx.
395.9
And gx is x squared, so that the answer is negative cosine of x squared plus C. So in the comments below this video, if you want to discuss, how could you use the reverse chain rule to answer this question? So how could you use the reverse chain rule to find the integral of 1 over 5x plus 6. If you want to, put your ideas in the comments below. Thank you for watching this video.

Watch the video for an explanation of integration, with a problem to practise solving yourself at the end.

## Discuss in the comments below:

How coud you use the reverse chain rule to find the integral of the example given in the video?