Determine the integral of nine sin 𝑥 plus four cos 𝑥.
To integrate this expression, what we can actually use is use two linearity rules to actually help us do this. The first of these linearity rules tells us that if we have the integral of 𝑓 𝑥 plus 𝑔 𝑥, then this is equal to the integral of 𝑓 𝑥 plus the integral of 𝑔 𝑥.
So what it means is we can actually split them up to find the integral. So we can do that with our expression. So we can say that the integral of nine sin 𝑥 plus four cos 𝑥 is gonna be equal to the integral of nine sin 𝑥 plus the integral of four cos 𝑥. Well, could we do any more simplifying before we actually integrate? And the answer is yes. We can actually use the second linearity rule.
And the second rule tells us that the integral of a constant multiplied by a function is equal to the constant multiplied by the integral of the function. And it is this because if we’re integrating, then the coefficient or the constant doesn’t affect the integral. Okay, so let’s use this and rewrite it once more.
So we now have an expression in the way that we can definitely go forward and integrate. So we’ve got nine multiplied by the integral of sin 𝑥 plus four multiplied by the integral of cos 𝑥. So what we can now use is we can now use our standard integrals to find the answer. Well, our first term is gonna be negative nine cos 𝑥. And we get that from one of our standard integrals, which tells us that the integral of sin 𝑥 is equal to negative cos 𝑥.
And then our second term is going to be positive four sin 𝑥. And we get this because the standard integral is that the integral of cos 𝑥 is equal to sin 𝑥. And then we have plus 𝑐 because we cannot forget our constant of integration. So therefore, we can say that the integral of nine sin 𝑥 plus four cos 𝑥 is equal to four sin 𝑥 minus nine cos 𝑥 plus 𝑐.