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The exercise contains eight integrals to solve. Let's work through them one by one.

### 1. \(\int_0^1 \frac{t^2 + 2t - 1}{t + 1} \, dt\)

We can attempt to simplify the expression before integrating by performing polynomial division of \(t^2 + 2t - 1\) by \(t + 1\).

### 2. \(\int_0^1 x(x^2 + 1)^5 \, dx\)

We can use substitution: let \(u = x^2 + 1\), and \(du = 2x \, dx\), transforming the integral into a simpler form.

### 3. \(\int_{-1}^4 \frac{dt}{(2t + 5)^3}\)

This can be solved using substitution: let \(u = 2t + 5\), and \(du = 2 \, dt\).

### 4. \(\int_{-4}^0 x \sqrt{x + 4} \, dx\)

This also lends itself to substitution: let \(u = x + 4\), and \(du = dx\).

### 5. \(\int_0^2 \frac{x}{\sqrt{2 - x + \sqrt{2 + x}}} \, dx\)

This one involves more advanced techniques, possibly substitution or special integrals, depending on how the square root behaves.

### 6. \(\int_2^4 \frac{dx}{x^2 - 1}\)

The denominator \(x^2 - 1\) can be factored into \((x - 1)(x + 1)\), and partial fraction decomposition can be used.

### 7. \(\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\sin(t)}{\cos^3(t)} \, dt\)

This can be simplified using trigonometric identities, for example, recognizing that \(\frac{\sin(t)}{\cos^3(t)} = \sec^3(t) \cdot \sin(t)\).

### 8. \(\int_0^{\frac{\pi}{2}} \sin^5(t) \, dt\)

We can use the power-reduction identity for \(\sin^5(t)\), which expresses powers of sine in terms of lower powers and cosines.

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Let me know which integrals you would like to start with, or if you prefer I solve all of them. Would you like any detailed explanation of the steps for a specific problem?

### 5 Related Questions:
1. How do you perform polynomial division for simplifying integrals?
2. What is the method of substitution in integral calculus?
3. How does partial fraction decomposition work for rational functions?
4. What are some useful trigonometric identities for integrals involving sine and cosine?
5. How can special functions or integrals, like elliptic integrals, appear in more complex problems?

### Tip:
When facing a complicated integral, always check if the integrand can be simplified by factoring, using identities, or making a substitution before proceeding directly with the integration.

Exercise 5 - Des intégrales plus ou moins faciles (ASR): Calculer les intégrales suivantes

This exercise focuses on solving integrals using techniques such as substitution, polynomial division, partial fraction decomposition, and trigonometric identities. Learn how to solve complex integrals step-by-step.

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