Let's solve these integrals step by step, **starting with (1.1):** --- ### **(1.1)** \[ \int \frac{2}{\sqrt{4 - x^2}} \, dx \] #### **Step 1: Recognize the Form** This looks like the derivative of \(\arcsin(x)\), which is: \[ \frac{d}{dx} \arcsin(x) = \frac{1}{\sqrt{1-x^2}} \] But we have \(4-x^2\) in the root. Let's rewrite \(4-x^2 = 2^2 - x^2\). #### **Step 2: Substitution** Let \(x = 2 \sin \theta\), so \(dx = 2 \cos \theta d\theta\). Now, \[ 4-x^2 = 4 - 4\sin^2\theta = 4\cos^2\theta \] So, \[ \sqrt{4-x^2} = 2\cos\theta \] #### **Step 3: Substitute into the Integral** \[ \int \frac{2}{\sqrt{4-x^2}} dx = \int \frac{2}{2\cos\theta} \cdot 2\cos\theta d\theta \] \[ = \int 2 d\theta \] #### **Step 4: Integrate** \[ \int 2 d\theta = 2\theta + C \] #### **Step 5: Back-Substitute \(\theta\)** Recall \(x = 2\sin\theta \implies \sin\theta = \frac{x}{2} \implies \theta = \arcsin\left(\frac{x}{2}\right)\) #### **Final Answer:** \[ \boxed{2\arcsin\left(\frac{x}{2}\right) + C} \] --- Would you like me to continue with the next integrals (e.g., 1.2, 1.3, etc.)? If yes, specify which ones or if you want all.