Video Transcript
Determine the integral of csc
squared of negative eight 𝑥 plus seven over seven with respect to 𝑥.
In this question, we begin by
noticing that the argument of our trigonometric function has the form 𝑎𝑥 plus
𝑏. We can rewrite negative eight 𝑥
plus seven over seven as negative eight-sevenths 𝑥 plus seven-sevenths, which
simplifies to negative eight-sevenths 𝑥 plus one. We will now perform integration by
substitution by first letting 𝑢 equal negative eight-sevenths 𝑥 plus one.
Differentiating, we have d𝑢 by d𝑥
is equal to negative eight-sevenths, which can be rearranged such that d𝑥 is equal
to negative seven-eighths d𝑢. Replacing negative eight 𝑥 plus
seven over seven with 𝑢 and d𝑥 with negative seven-eighths d𝑢, we have the
integral of csc squared 𝑢 multiplied by negative seven-eighths d𝑢. We can then factor out the constant
negative seven-eighths, giving us negative seven-eighths multiplied by the integral
of csc squared 𝑢 d𝑢.
Next, we recall the standard result
for the indefinite integral of the square of the cosecant function. The integral of csc squared 𝑥 with
respect to 𝑥 is equal to negative cot 𝑥 plus 𝐶. Applying this to our question gives
us negative negative seven-eighths cot 𝑢 plus 𝐶, which in turn simplifies to
seven-eighths cot 𝑢 plus 𝐶. Finally, we can replace 𝑢 with
negative eight 𝑥 plus seven over seven such that the integral of csc squared
negative eight 𝑥 plus seven over seven with respect to 𝑥 is equal to seven-eighths
cot of negative eight 𝑥 plus seven over seven plus 𝐶.
