(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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