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When solving a quadratic equation like this it is useful to write it in the form (x+a)(x+b)=0, as this is simply saying 'two numbers multiplied together equal zero'. A general rule in maths is that whenever two numbers multiply together to equal zero, either one or both of those numbers equal zero (try multiplying any number you can think of by zero and see what you get!)
The next question is how do I write x^2+4x+3 in the form (x+a)(x+b)? The answer to this is simple, you need to search for two numbers which multiply together to make three, and also add together to make 4. In this case the answer is 3 and 1, so we can re-write our original equation as (x+3)(x+1)=0. Now because this product is equal to zero, we can write that either x+3=0 or x+1=0 (because when a product equals zero either one or both of the numbers being multiplied must be zero).
Starting with x+3=0, subtracting 3 from both sides gives us our first solution: x=-3.
We can now subtract 1 from both sides of x+1=0, giving us our second solution: x=-1.
As you can see, there are two solutions to this equation! If you find this hard to believe, you can check that both solutions are correct by replacing all of the x's in the original equation with each solution in turn, and they should both equal zero!see more
At first, you might think that it is possible to perform this integral simply by inspection, using the 'backwards chain rule'. This method would consist of adding one to the power, to get cos3(x), then dividing by the new power and the derivative of the function, giving you -(1/3sin(x))cos3(x). However, once you have performed an integration it is always wise to check your result by differentiating to see if you get your starting function back. In this case, it is clear that differentiating -(1/3sin(x))cos3(x) does not give cos2(x), because you have to use the quotient rule to differentiate cos3(x)/sin(x).
This means that a different approach is required to perform the integration, and that is to use the trig identity cos2x=1/2+(1/2)cos(2x) to change the integrand to something which can be integrated easily. It is then simple to integrate 1/2 +(1/2)cos(2x) using the familiar method, giving the correct answer of (1/2)x+(1/4)sin(2x)+c (not forgetting the constant of integration!).
Similarly, sin2(x) can be integrated quickly using the trig identity sin2(x)=1/2-(1/2)cos(2x), so these two identities are definitely worth memorizing!see more