Video Transcript
Determine the indefinite integral
of negative sin 𝑥 minus nine times the cos of 𝑥 evaluated with respect to 𝑥.
Before trying to evaluate this, it
can be useful to recall some of the properties of integrals. Firstly, the integral of the sum of
two or more functions is 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. These properties mean 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.
And now we recall the general
results for the integrals of the sine and cosine functions. The indefinite integral of sin of
𝑎𝑥 is equal to negative one over 𝑎 times cos 𝑎𝑥 plus the constant of
integration 𝑐. And the integral of cos of 𝑎𝑥
evaluated with respect to 𝑥 is equal to one over 𝑎 times sin 𝑎𝑥 plus the
constant 𝑐. So in our case, the constant 𝑎 is
equal to one, and our integral is negative negative cos 𝑥 plus the constant 𝐴
minus nine times sin 𝑥 plus the constant 𝐵. And we’ve chosen 𝐴 and 𝐵 to show
that these are different constants of integration.
Now distributing the parentheses
and combining the two constants 𝐴 and 𝐵 into a single constant 𝐶, we have the
indefinite integral of negative sin 𝑥 minus nine cos 𝑥 evaluated with respect to
𝑥 is equal to cos 𝑥 minus nine times sin 𝑥 plus the constant 𝐶.
