Video Transcript
Determine the indefinite integral
of negative five multiplied by the cot squared of four 𝑥 plus seven plus one with
respect to 𝑥.
To begin, since negative five is a
constant, we can take it outside the integrand. Next, since the argument of the
trigonometric function is four 𝑥 plus seven instead of 𝑥, we can make a
substitution. We can let 𝑢 equal four 𝑥 plus
seven. And it then follows that d𝑢 by d𝑥
is equal to four or equivalently one-quarter d𝑢 is equal to d𝑥. Making this change of variable in
the integral, we obtain negative five multiplied by the indefinite integral of cot
squared 𝑢 plus one one-quarter d𝑢, which we can write as negative five over four
multiplied by the indefinite integral of cot squared 𝑢 plus one with respect to
𝑢.
The integrand contains the square
of the cotangent function. The antiderivative of cot squared
𝑥 is not readily available, but we do know that cot squared 𝑥 can be expressed in
terms of csc squared 𝑥, which we do know the antiderivative of. We recall that cot squared of 𝑥 is
equal to csc squared of 𝑥 minus one. And this can be obtained from the
Pythagorean identity. Hence, the integrand becomes csc
squared of 𝑢 minus one plus one or simply csc squared of 𝑢. We can now proceed by recalling the
standard result that the indefinite integral of csc squared of 𝑥 with respect to 𝑥
is equal to negative cot of 𝑥 plus 𝐶.
Applying this result, we obtain
negative five over four multiplied by negative cot 𝑢 plus 𝐶. Reversing our substitution, so
replacing 𝑢 with four 𝑥 plus seven, we obtain our final answer, which is that the
indefinite integral of negative five multiplied by cot squared of four 𝑥 plus seven
plus one with respect to 𝑥 is equal to five over four multiplied by cot of four 𝑥
plus seven plus 𝐶.
