Video Transcript
Determine the indefinite integral
of negative eight sin of eight π₯ minus seven cos of five π₯ evaluated with respect
to π₯.
In this question, weβre looking to
integrate the sum of two functions in π₯. We begin by recalling the fact that
the integral of the sum of two or more functions is actually equal to the sum of the
integrals of those respective functions. So we can write our problem as the
integral of negative eight sin of eight π₯ with respect to π₯ plus the integral of
negative seven cos of five π₯ dπ₯. We also know that we can take any
constant factors outside of the integral and focus on integrating each expression in
π₯. So we can rewrite our problem
further as negative eight times the integral of sin of eight π₯ with respect to π₯
minus seven times the integral of cos of five π₯ with respect to π₯.
Next, we recall the general results
for the integral of sine and cosine. The integral of sin of ππ₯ is
negative one over π cos of ππ₯ plus π. And the indefinite integral of cos
of ππ₯ is one over π sin of ππ₯ plus π. We integrate each function,
respectively, and we see that the integral of sin of eight π₯ is negative
one-eighths cos of eight π₯ plus π΄. And the indefinite integral of cos
of five π₯ is a fifth sin of five π₯ plus π΅. And Iβve chosen π΄ and π΅, as
opposed to just one value of π, to show that these are actually different
constants.
Our final step is to distribute the
parentheses. Negative eight times negative
one-eighths cos of eight π₯ is just cos of eight π₯. Negative seven times one-fifth of
sin of five π₯ is a negative seven-fifths sin of five π₯. Finally, we multiply negative eight
by π΄ and negative seven by π΅. And we end up with this new
constant πΆ. And we found that the integral we
required is cos of eight π₯ minus seven-fifths of sin of five π₯ plus πΆ.
