Determine the indefinite integral
of negative sin of 𝑥 minus nine cos of 𝑥 evaluated with respect to 𝑥.
It can be useful to recall the
properties of integrals before evaluating this. Firstly, we know that the integral
of the sum of two or more functions is actually equal to the sum of the integrals of
those respective functions. And we also know that we can take
any constant factors outside of the integral and focus on integrating the expression
in 𝑥 itself. This means we can rewrite our
integral as negative the integral of sin 𝑥 evaluated with respect to 𝑥 minus nine
times the integral of cos of 𝑥 evaluated with respect to 𝑥. This means we can rewrite our
integral as shown. And this means we can recall the
general results for the integral of the sine and cosine functions.
The indefinite integral of sin of
𝑎𝑥 is negative one over 𝑎 cos 𝑎𝑥 plus that constant 𝑐. And the indefinite integral of cos
of 𝑎𝑥 is one over 𝑎 times sin of 𝑎𝑥 plus 𝑐. So our integral is negative
negative cos of 𝑥 plus 𝐴 minus nine times sin of 𝑥 plus 𝐵. Then I’ve chosen 𝐴 and 𝐵 here to
show that these are different constants of integration. Distributing the parentheses and
combining our constants in to one constant, we find the integral of negative sin 𝑥
minus nine cos of 𝑥 evaluated with respect to 𝑥 to be cos of 𝑥 minus nine sin of
𝑥 plus 𝑐.