Integral Simplification Using Substitution

Perform the substitution $u = x + 1$ and express the integral $\displaystyle \int \sqrt{1 - (x + 1)^2}^5 \, dx$ in terms of $u$.

In calculus, the technique of substitution is a powerful tool for simplifying integrals by changing the variable of integration. This process, also known as u-substitution, involves substituting a portion of the integral with a new variable, typically denoted as $u$. This not only simplifies the integration process but also helps to address more complex integrals that involve compositions of functions or difficult algebraic expressions.

In this problem, the substitution $u = x + 1$ is suggested. The goal is to simplify the integral $extstyle \\int \\sqrt{1 - (x + 1)^2}^5 \, dx$ by expressing it in terms of the new variable $u$. This specific substitution aids in transforming the given integral into a form that is easier to manage, often converting the expression into a standard integral form. Such transformations are crucial when the integrand involves complicated expressions, like compositions or powers of trigonometric, exponential, or logarithmic functions, which can be daunting to handle directly.

As you substitute and integrate, you'll find that this technique not only enhances your ability to simplify complex expressions but also deepens your understanding of how functions behave under transformation. This approach is especially valuable in scenarios involving integrals that arise in various applications of calculus, including physics and engineering problems where simplifying complex systems is a frequent necessity. Understanding and mastering substitution is thus essential for tackling a wide range of integrals efficiently.

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