Video Transcript
Determine the integral of 48 minus six ๐ฅ over the fifth root of 16 minus two ๐ฅ d๐ฅ.
Itโs not instantly obvious how we need to evaluate this integral. However, if we look carefully, we see that the numerator is a scalar multiple of the inner function on the denominator such that 48 minus six ๐ฅ is three times 16 minus two ๐ฅ. This is a hint to us that weโre going to need to use integration by substitution to evaluate this indefinite integral.
Weโre going to let ๐ข be equal to 16 minus two ๐ฅ. Weโve chosen this part for our substitution as 16 minus two ๐ฅ is the inner function in a composite function. We differentiate ๐ข with respect to ๐ฅ, and we see that d๐ข by d๐ฅ is equal to negative two. Now, remember, d๐ข by d๐ฅ is not a fraction, but we treat it a little like one when performing integration by substitution. And we can see that this is equivalent to saying negative one-half d๐ข is equal to d๐ฅ.
Letโs substitute what we now have into our original integral. We saw that 48 minus six ๐ฅ is equal to three times 16 minus two ๐ฅ. So, the numerator becomes three ๐ข. The denominator becomes the fifth root of ๐ข. And we replace d๐ฅ with negative a half d๐ข. Letโs take out a factor of negative three over two. And weโll write our denominator as ๐ข to the fifth power. Weโre dividing ๐ข to the power of one by ๐ข to the power of one-fifth.
So, we subtract one-fifth from one, and weโre left with ๐ข to the four-fifths. The antiderivative of ๐ข to the four-fifths is ๐ข to the nine-fifths divided by nine-fifths. Thatโs the same as five-ninths times ๐ข to the nine-fifths. And, remember, this is an indefinite integral, so we add that constant of integration ๐.
When we distribute the parentheses, we have negative five-sixths times ๐ข to the nine-fifths plus capital ๐ถ. Since our original constant has been multiplied by negative three over two, and since we were evaluating an integral in terms of ๐ฅ, we must replace ๐ข with 16 minus two ๐ฅ. And we see that our integral is negative five-sixths times 16 minus two ๐ฅ to the nine-fifths plus ๐ถ.