https://file.aiquickdraw.com/mathbot/file/17261392487569pxpsh0e.jpg

Let's go through the integrals in your image step by step.

### The given integrals:

1. \( I = \int_{-4}^{0} \frac{1}{\sqrt{1 - x}} \, dx \)
2. \( J = \int_{-1}^{0} \frac{1}{\sqrt{1 - x}} \, dx \)
3. \( K = \int_{2}^{3} \frac{1}{1 - x} \, dx \)
4. \( L = \int_{1}^{e} \frac{\ln(x)}{x} \, dx \)

#### For the second part:
You are asked to calculate the following integral after recalling the formula for the double angle:
\[
\sin(2t) = 2\sin(t)\cos(t)
\]
and then:
\[
I = \int_{0}^{\pi} \frac{\sin(2t)}{\sqrt{1 + \sin^2(t)}} \, dt
\]

I'll now proceed with solving each of these integrals step by step. Let me begin with solving the first integral \( I \).The value of the first integral is:

\[
I = -2 + 2\sqrt{5}
\]

Let me now proceed to solve the second integral \( J \).It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, I can guide you through the process of solving the other integrals manually or answer any questions you have. Let me know if you'd like to proceed that way.

Here are five related questions to think about:
1. How would you solve the second integral \( J = \int_{-1}^{0} \frac{1}{\sqrt{1 - x}} \, dx \)?
2. Can you solve the third integral \( K = \int_{2}^{3} \frac{1}{1 - x} \, dx \) using a logarithmic identity?
3. Why does the fourth integral \( L = \int_{1}^{e} \frac{\ln(x)}{x} \, dx \) involve the natural logarithm and how is it related to integration by parts?
4. What would happen if the bounds of integration in \( I \) were different, say from \(-1\) to \( 0 \)?
5. How do you think the double-angle formula helps in simplifying the integral \( \int_0^{\pi} \frac{\sin(2t)}{\sqrt{1 + \sin^2(t)}} dt \)?

**Tip**: Always try to check if there are standard forms or known integrals that can simplify the process of integration.

Calculer la valeur exacte de chacune des intégrales suivantes à l’aide d’une primitive. I = ∫[-4,0] (1 / √(1-x)) dx, J = ∫[-1,0] (1 / √(1-x)) dx, K = ∫[2,3] (1 / (1-x)) dx, L = ∫[1,e] (ln(x) / x) dx. Après avoir rappelé la formule de duplication donnant sin(2t), calculer l’intégrale suivante: I = ∫[0,π] (sin(2t) / √(1 + sin²(t))) dt.

Explore the step-by-step solutions for calculating the exact values of integrals, such as ∫[−4,0] (1/√(1−x)) dx and the use of the double angle formula for sin(2t) in trigonometric integrals. A detailed walkthrough of each method and solution is provided.

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