# Video: Finding the General Antiderivative of a Function Using the Power Rule of Integration with Fractional Exponents

Determine ∫ (−4 ^(5)√(𝑥⁹) + 8) ^(5)√(𝑥²) d𝑥.

02:10

### Video Transcript

Determine the indefinite integral having negative four multiplied by the fifth root of 𝑥 to the power of nine plus eight all multiplied by the fifth root of 𝑥 squared with respect to 𝑥.

Let’s start by writing this roots as powered. We know that the 𝑛th root of 𝑥 is equal to 𝑥 to the power of one over 𝑛. Once we’ve written our roots as powers, we can then combine them with the existing powers, using the fact that 𝑥 to the power of 𝑛 to the power of 𝑚 is equal to 𝑥 to the power of 𝑛 multiplied by 𝑚. Therefore, 𝑥 to the power of nine to the power of one-fifth becomes 𝑥 to the power of nine-fifths. And 𝑥 squared to the power of one-fifths becomes 𝑥 to the power of two-fifths. Now, we can expand the brackets, using the fact that 𝑥 to the power of 𝑛 times 𝑥 to the power of 𝑚 is equal to 𝑥 to the power of 𝑛 plus 𝑚. So our integral becomes the integral of negative four 𝑥 to the power of eleven-fifths plus eight multiplied by 𝑥 to the power of two-fifths with respect to 𝑥.

Here, we can use the power rule for integration which tells us that the indefinite integral of 𝑥 to the power of 𝑛 with respect to 𝑥 is equal to 𝑥 to the power of 𝑛 plus one over 𝑛 plus one plus 𝐶. We can apply this rule to our integral term by term. For the first term, we have negative four 𝑥 to the power of eleven-fifths. Therefore, 𝑛 is equal to eleven-fifths. When we integrate this term, we get negative four multiplied by 𝑥 to the power of 𝑛 plus one. And 𝑛 plus one is simply sixteen-fifths. So it’s 𝑥 to the power of sixteen-fifths. And then we need to divide by 𝑚 plus one. So that’s dividing by sixteen-fifths.

For the second time, we have eight multiplied by 𝑥 to the power of two-fifths. Therefore, 𝑛 is two-fifths. So we add eight multiplied by 𝑥 to the power of 𝑛 plus one which is 𝑥 to the power of seven-fifths. And we then divide by seven-fifths. And we mustn’t forget to add our constant of integration 𝐶. Now, all that remains to do is to simplify. And so we obtain a solution that the indefinite integral of negative four multiplied by the fifth root of 𝑥 to the power of nine plus eight all multiplied by the fifth root of 𝑥 squared with respect to 𝑥 is equal to negative five multiplied by 𝑥 to the power of sixteen-fifths over four plus 40 multiplied by 𝑥 to the power of seven-fifths over seven plus 𝐶.