What is the result of the integral ∫ x^n dx, where n ≠ -1?

What is the result of the integral ∫ x^n dx, where n ≠ -1?

• A. (x^(n+1))/(n+1) + C
• B. (x^(n-1))/(n-1) + C
• C. n*x^(n-1) + C
• D. (1/n)*x^n + C

Answer: (A)

The result of the integral ∫ x^n dx, where n is not equal to -1, is determined by applying the power rule for integration. This rule states that when integrating a function of the form x^n, you increase the exponent by 1 and then divide by the new exponent. Therefore, when you integrate x^n, you add 1 to the exponent n, resulting in n + 1. The integral can then be expressed as: ∫ x^n dx = (x^(n+1)) / (n + 1) + C Here, C represents the constant of integration, which is necessary when performing indefinite integrals since there are infinitely many antiderivatives differing by a constant. This formulation specifically applies since n is not equal to -1; if n were -1, the integral would represent a logarithmic function instead due to the undefined behavior of division by zero. The correct answer encapsulates this principle accurately, confirming that the integral of x raised to the power n (where n is not -1) is equal to (x^(n + 1)) / (n + 1) plus the integration constant C. Other provided options do not align with this fundamental integration process,

The result of the integral ∫ x^n dx, where n is not equal to -1, is determined by applying the power rule for integration. This rule states that when integrating a function of the form x^n, you increase the exponent by 1 and then divide by the new exponent.

Therefore, when you integrate x^n, you add 1 to the exponent n, resulting in n + 1. The integral can then be expressed as:

∫ x^n dx = (x^(n+1)) / (n + 1) + C

Here, C represents the constant of integration, which is necessary when performing indefinite integrals since there are infinitely many antiderivatives differing by a constant.

This formulation specifically applies since n is not equal to -1; if n were -1, the integral would represent a logarithmic function instead due to the undefined behavior of division by zero.

The correct answer encapsulates this principle accurately, confirming that the integral of x raised to the power n (where n is not -1) is equal to (x^(n + 1)) / (n + 1) plus the integration constant C. Other provided options do not align with this fundamental integration process,