Question

Evaluate the indefinite integral.

∫(x^4 - x^5 + 5) dx
a. (1/5)x^5 - (1/6)x^6 + 5x + C
b. (1/5)x^5 + (1/6)x^6 - 5x + C
c. (1/5)x^5 + (1/6)x^6 + 5x + C
d. (1/5)x^5 - (1/6)x^6 - 5x + C

Answer 1

The indefinite integral a.
`\((1)/(5)x^5 - (1)/(6)x^6 + 5x + C\)`

Therefore, correct option is a.
`\((1)/(5)x^5 - (1)/(6)x^6 + 5x + C\)`

Step-by-step explanation:

To evaluate the indefinite integral
`\(\int(x^4 - x^5 + 5) \,dx\)`, apply the power rule of integration. For each term, add 1 to the exponent and divide by the new exponent.

The result is
`\((1)/(5)x^5 - (1)/(6)x^6 + 5x + C\)`, where C is the constant of integration.

This process ensures that the antiderivative of each term in the integrand is correctly computed. The correct option is a, as it precisely matches the result obtained through the integration process.

Therefore, correct option is a.
`\((1)/(5)x^5 - (1)/(6)x^6 + 5x + C\)`

Answer 2

Apply the power rule for integration to each term:
`∫x^n dx = (1/(n+1))x^(n+1)`. This yields
`(1/5)x^5 - (1/6)x^6 + 5x + C`, matching option (a).

Step-by-step explanation:

The given indefinite integral
`∫(x^4 - x^5 + 5)` dx can be solved by applying the power rule for integration, which states that
`∫x^n dx = (1/(n+1))x^(n+1)`. Applying this rule to each term individually, the integral becomes
`(1/5)x^5 - (1/6)x^6 + 5x + C`, where C is the constant of integration. In the first term,
`x^4` integrates to
`(1/5)x^5`, in the second term,
`x^5` integrates to
`-(1/6)x^6`, and in the third term, the integral of the constant term 5 is 5x. The constant of integration, denoted by C, represents an arbitrary constant that can take any real value.

Option (a)
`(1/5)x^5 - (1/6)x^6 + 5x + C` perfectly aligns with the derived result, confirming it as the correct answer. The power rule is a fundamental technique in calculus, establishing a direct relationship between derivatives and integrals for power functions.

Understanding and applying such rules are crucial for solving a wide range of integration problems efficiently, providing a systematic approach to finding antiderivatives.