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The formula for the integral of x^n is (x^(n+1))/(n+1) + C, where C is the constant of integration.
To prove this formula, we can use the power rule of integration. According to this rule, the integral of x^n is (x^(n+1))/(n+1) + C, where C is the constant of integration. We can verify this formula by differentiating it and checking if we get back the original function.
Let's differentiate (x^(n+1))/(n+1) + C with respect to x. Using the power rule of differentiation, we get:
d/dx [(x^(n+1))/(n+1) + C] = (n+1)x^n
This is the same as the original function x^n, except for the constant factor (n+1). Therefore, we can conclude that the formula (x^(n+1))/(n+1) + C is indeed the antiderivative of x^n.
Another way to prove this formula is to use integration by substitution. Let u = x^(n+1), then du/dx = (n+1)x^n. Rearranging, we get dx = du/((n+1)x^n). Substituting these into the integral of x^n, we get:
∫x^n dx = ∫u du/((n+1)x^n)
= (1/(n+1)) ∫u^(-n/(n+1)) du
= (1/(n+1)) u^((n+1)/(n+1)) + C
= (x^(n+1))/(n+1) + C
This is the same as the formula we derived earlier using the power rule of integration. Therefore, we have proven the formula for the integral of x^n.
For further understanding, you can explore various integration techniques
, which can be applied to different types of functions. Additionally, the concepts of definite and indefinite integrals
are foundational to understanding the broader context of integration in calculus. For a solid basis on how these rules are applied, reviewing the basic integration rules
can also be beneficial.
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