Question

Int 1[(x - 1)^3(x + 2)^5]^1/4 dx is equal to

Answer 1

The integral ∫
`[(x - 1)^3(x + 2)^5]^1^/^4` dx is best solved using the substitution method, resulting in the expression (1/5) * ∫
`u^(^1^/^4^)` du, where u =
`(x - 1)^3(x + 2)^5`.

Step-by-step explanation:

To solve the given integral, we can use the substitution method. Let u =
`(x - 1)^3(x + 2)^5`. Then, differentiate both sides with respect to x to find du/dx. This allows us to express dx in terms of du.

Now, substitute u and the expression for dx back into the original integral. The integral becomes ∫
`u^(^1^/^4^)` * (1/5) du. Integrate with respect to u, and then substitute back for u to find the final result.

This approach simplifies the integration process by transforming the original expression into a more manageable form. Substitution is a powerful technique in calculus, particularly useful for handling complex expressions like the one given in the question.

Integration techniques, specifically the substitution method, play a crucial role in simplifying complex integrals. Understanding when and how to apply substitution can greatly enhance problem-solving skills in calculus, providing a systematic approach to tackle intricate mathematical expressions.