Choosing a Formula In Exercises 73-76, state the integration...

Choosing a Formula In Exercises $73-76,$ state the integration formula you would use to perform the integration. Explain why you chose that formula. Do not integrate.
$$
\int x\left(x^{2}+1\right)^{3} d x
$$

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Key Concepts

Integration by Substitution

This technique involves substituting a part of the integrand with a new variable (usually labeled u) to simplify the integral. The choice of u is typically guided by identifying an inner function whose derivative appears elsewhere in the integrand. This method effectively reverses the chain rule from differentiation, allowing for a streamlined computation once the variable change is performed.

Reverse Chain Rule

The reverse chain rule is used when you recognize that an integral is the result of a derivative computed using the chain rule. In these cases, setting u equal to an inner function transforms the differential into a multiple of du. This approach directly leads to an integral in terms of u, which is typically easier to handle due to its simpler structure.

Power Rule for Integration

After applying substitution, many integrals reduce to a form where the integrand is a power of the new variable, u. The power rule for integration states that ?u^n du = u^(n+1)/(n+1) + C, provided n is not -1. This rule is then used to integrate the simplified expression, completing the process initiated by the substitution.

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