To solve the integral

$\int_0^1 \left(x^2 + 1\right)^{10} \cdot (2x) \, \mathrm{d}x,$

we can use substitution. Let's proceed step-by-step.

Step 1: Substitution

Let $u = x^2 + 1.$ Then, the differential $\mathrm{d}u$ is found by differentiating $u$ with respect to $x$: $\frac{\mathrm{d}u}{\mathrm{d}x} = 2x \Rightarrow \mathrm{d}u = 2x \, \mathrm{d}x.$

This matches the $2x \, \mathrm{d}x$ in the integrand, so we can substitute $\left(x^2 + 1\right)^{10} \cdot (2x) \, \mathrm{d}x$ with $u^{10} \, \mathrm{d}u$.

Step 2: Change of Limits

When $x = 0$: $u = 0^2 + 1 = 1.$ When $x = 1$: $u = 1^2 + 1 = 2.$

Step 3: Substitute and Integrate

The integral becomes $\int_1^2 u^{10} \, \mathrm{d}u.$ Now, integrate $u^{10}$: $\int u^{10} \, \mathrm{d}u = \frac{u^{11}}{11}.$

Step 4: Apply Limits

Now substitute the limits $u = 1$ and $u = 2$: $\left. \frac{u^{11}}{11} \right|_1^2 = \frac{2^{11}}{11} - \frac{1^{11}}{11}.$

Calculating this gives: $= \frac{2^{11} - 1}{11} = \frac{2048 - 1}{11} = \frac{2047}{11}.$

Final Answer

$\int_0^1 \left(x^2 + 1\right)^{10} \cdot (2x) \, \mathrm{d}x = \frac{2047}{11}.$

Would you like more details or have any questions?

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Here are some related questions you might explore:

1. How would the solution change if the power were different, say $\left(x^2 + 1\right)^n$?
2. What substitution would work if the integrand were $\left(x^3 + 1\right)^{10}(3x^2)$?
3. How does changing the limits affect the integral's outcome?
4. What would the integral look like if it extended from $x = 0$ to $x = 2$?
5. How can substitution be applied to more complex integrands?

Tip: When you spot a term with $f(x)$ and $f'(x)$ in the integrand, substitution is often a powerful technique to simplify the problem.