Video Transcript
Determine the indefinite integral
with respect to 𝑥 of negative eight times the sin of eight 𝑥 minus seven times the
cos of five 𝑥.
In this question, we’re looking to
integrate the sum of two functions of 𝑥. We begin by recalling that the
integral of the sum of two or more functions is equal to the sum of the integrals of
those respective functions. And so we can write our integral as
the integral of negative eight sin of eight 𝑥 with respect to 𝑥 plus the integral
of negative seven cos of five 𝑥 d𝑥. We also know that we can take any
constant factors outside of the integrals and focus on integrating each expression
in 𝑥. And this means we can rewrite our
integrals as negative eight times the integral with respect to 𝑥 of sin eight 𝑥
minus seven times the integral of cos five 𝑥 d𝑥.
Next, we recall the general results
for the integrals of sin 𝑎𝑥 and cos 𝑎𝑥. The indefinite integral with
respect to 𝑥 of sin of 𝑎𝑥 is equal to negative one over 𝑎 times cos 𝑎𝑥 plus
the constant of integration 𝑐. And the indefinite integral of cos
𝑎𝑥 with respect to 𝑥 is one over 𝑎 sin 𝑎𝑥 plus the constant 𝑐. In our case, the constant 𝑎 is
eight in the first integral and five in the second integral. And applying these results to our
integrals, we see that the integral of sin of eight 𝑥 is negative one over eight
cos eight 𝑥 plus the constant 𝐴. And the integral with respect to 𝑥
of cos of five 𝑥 is one over five times the sin of five 𝑥 plus 𝐵. Note that we’ve chosen 𝐴 and 𝐵 as
the constant of integration, not just a single value of 𝑐, to show that these are
actually different constants.
Our final step is to distribute the
parentheses. Negative eight times negative one
over eight. cos of eight 𝑥 is just the cos of
eight 𝑥. And negative seven times one over
five sin five 𝑥 is negative seven over five sin five 𝑥. And finally, we multiply negative
eight by 𝐴 and negative seven by 𝐵. And since we don’t know the values
of 𝐴 and 𝐵, we choose to represent this as a single constant 𝑐. We found then that the integral we
required is the cos of eight 𝑥 minus seven over five times the sin of five 𝑥 plus
the constant 𝑐.
