Master the Power Rule of Integration: Indefinite Integrals of x^(n+1)/(n+1)

(x^(n+1)/n+1) + C

∫x^n dx

This is the indefinite integral of x^(n+1)/(n+1). Here, C represents the constant of integration.

To understand how this result was obtained, we need to use the power rule of integration. The power rule states that the integral of x^n is (x^(n+1))/(n+1), provided n is not equal to -1.

Using the power rule, we can integrate x^(n+1) term as follows:

∫x^(n+1) dx = x^(n+2)/(n+2), (using the power rule of integration)

Now, this result can be modified to the given integral by dividing by (n+1),

∫( x^(n+1)/(n+1) ) dx = ( x^(n+2)/(n+2) )/(n+1) + C

Multiplying the numerator and denominator by (n+2), we can simplify it as:

∫( x^(n+1)/(n+1) ) dx = x^(n+2)/(n+2)(n+1) + C

This is the required result, which can also be written as:

∫( x^(n+1)/(n+1) ) dx = (1/(n+1)) * x^(n+1) + C

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