QuestionFind the most general antiderivative of the function
Answer: (i)
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STEP 1
1. We are given the function g(x) = √[7](x^6) + √[6](x^7).
2. We need to find the most general antiderivative G(x).
3. The most general antiderivative includes an arbitrary constant C.
STEP 2
1. Simplify the given function if possible.
2. Find the antiderivative of the first term.
3. Find the antiderivative of the second term.
4. Combine the results and add the arbitrary constant.
STEP 3
Let's examine the given function:
g(x) = √[7](x^6) + √[6](x^7)
We can simplify this by rewriting the roots as fractional exponents:
g(x) = x^(6/7) + x^(7/6)
STEP 4
Now, let's find the antiderivative of the first term, x^(6/7):
STEP 5
Next, let's find the antiderivative of the second term, x^(7/6):
STEP 6
Now, we combine the results from steps 2 and 3, and add a single arbitrary constant C (which absorbs C_1 and C_2):
This is the most general antiderivative of g(x).
Therefore, the most general antiderivative G(x) is:
This expression should be entered into the square (i) in your answer.
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