Evaluate the integral ∫ (5x^4) dx.

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Question: Evaluate the integral ∫ (5x^4) dx.

Options:

Correct Answer: x^5 + C

Solution:

The integral is (5/5)x^5 + C = x^5 + C.

Evaluate the integral ∫ (5x^4) dx.

Evaluate the integral ∫ (5x^4) dx.

Step 1: Identify the function to integrate, which is 5x^4.
Step 2: Use the power rule for integration. The power rule states that ∫x^n dx = (1/(n+1))x^(n+1) + C, where n is the exponent.
Step 3: In our case, n is 4. So, we apply the power rule: ∫5x^4 dx = 5 * (1/(4+1))x^(4+1) + C.
Step 4: Calculate (4+1) which equals 5. Now we have: ∫5x^4 dx = 5 * (1/5)x^5 + C.
Step 5: Simplify the expression. The 5 in the numerator and the 5 in the denominator cancel out: ∫5x^4 dx = x^5 + C.
Step 6: Write the final answer: x^5 + C.

Integration of Polynomials – The process of finding the antiderivative of a polynomial function, applying the power rule for integration.