In this problem, the task is to determine an appropriate substitution for the integral

$\int_0^1 10x \cos (5x^2 + 1) \, dx$

without evaluating it. Here’s a breakdown of the solution provided in the image:

1. Substitution Selection: The substitution chosen is $u = 5x^2 + 1$. This is appropriate because the integrand contains $5x^2 + 1$ within the cosine function, and differentiating $u$ with respect to $x$ will yield a term involving $x$, allowing us to simplify the integral.

2. Finding $du$: Differentiating $u$ with respect to $x$, $du = 10x \, dx.$ This directly matches the $10x \, dx$ part of the original integrand, which means $dx$ can be replaced by $\frac{du}{10x}$, simplifying the expression.

3. Adjusting the Limits: Since the limits of integration are in terms of $x$, we need to change them to be in terms of $u$.

   • When $x = 0$: $u = 5(0)^2 + 1 = 1$.
   • When $x = 1$: $u = 5(1)^2 + 1 = 6$.

4. Substituted Integral: Rewriting the integral in terms of $u$, $\int_0^1 10x \cos (5x^2 + 1) \, dx = \int_1^6 \cos(u) \, du.$

The transformed integral is now $\int_1^6 \cos(u) \, du$, which is simpler to evaluate, although the instructions specify not to evaluate it.

Would you like further details on this substitution method or have any questions about this problem?

Here are five related questions that build on this concept:

1. How do we decide the best substitution for an integral involving trigonometric functions?
2. What other techniques, besides substitution, are useful for integrals involving products like $10x \cos (5x^2 + 1)$?
3. What would the next steps be if we were to evaluate the integral $\int_1^6 \cos(u) \, du$?
4. Why is it important to change the limits of integration when performing substitution on definite integrals?
5. How does the chain rule relate to substitution in integral calculus?

Tip: When performing a substitution in a definite integral, always remember to update the limits of integration to reflect the new variable; this avoids the need to back-substitute at the end.