∫x^n dx
The integral of x raised to the power of n, denoted as ∫x^n dx, represents the antiderivative or the indefinite integral of the given function
The integral of x raised to the power of n, denoted as ∫x^n dx, represents the antiderivative or the indefinite integral of the given function. To evaluate this integral, we need to know the value of n.
Case 1: When n is not equal to -1:
If n is any real number other than -1, we can use the Power Rule of Integration. According to the power rule, the indefinite integral of x^n with respect to x is given by:
∫x^n dx = (x^(n+1))/(n+1) + C
Here, C represents the constant of integration.
For example:
If we have the integral of x^2 dx, we can use the power rule to find:
∫x^2 dx = (x^(2+1))/(2+1) + C
= (x^3)/3 + C
Case 2: When n is equal to -1:
If n is -1, the integral of x^(-1) is a special case and requires a different approach. The integral of x^(-1) can be calculated as the natural logarithm (ln) of the absolute value of x, plus a constant of integration.
∫x^(-1) dx = ln|x| + C
For example:
If we have the integral of x^(-1) dx, we can find:
∫x^(-1) dx = ln|x| + C
It’s important to note that when evaluating integrals, the constant of integration (C) should always be added, as it represents an infinite number of possible constant values that would still give the same derivative.
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