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Chapter 4: Problem 11

Find all the antiderivatives of the following functions. Check your work by taking derivatives. $$f(x)=5 x^{4}$$

Short Answer

Expert verified
Question: Find all the antiderivatives of the function f(x) = 5x^4. Answer: The antiderivatives of the function f(x) = 5x^4 are F(x) = x^5 + C, where C is a constant.

Step by step solution

01

Identify the given function

The given function is: $$f(x) = 5x^4$$
02

Integrate the function

To find the antiderivative, integrate f(x) with respect to x: $$\int f(x) dx = \int 5x^4 dx$$ Apply the power rule for integration: \(\int x^n dx = \frac{1}{n + 1} x^{n + 1} + C\), where n is a constant and C is the constant of integration. $$\int 5x^4 dx = 5 \int x^4 dx = 5\frac{1}{4+1}x^{4+1} + C = x^5 + C$$
03

Verify the antiderivative

To check our work, differentiate the antiderivative with respect to x: $$\frac{d}{dx}(x^5 + C)$$ Apply the power rule for differentiation: \(\frac{d}{dx}(x^n) = nx^{n-1}\). $$\frac{d}{dx}(x^5 + C) = 5x^{5-1} = 5x^4$$ This is the same as the original function, so our antiderivative is correct.
04

Write the final answer

The antiderivatives of the function f(x) = 5x^4 are: $$F(x) = x^5 + C$$

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