2) $$ \int(2 x-5)^{4} d x= $$ 3) $$ \int \cos x e^{\sin x} d x= $$ 1

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UpStudy AI · Accepted Answer

Sure, let's solve each integral step by step.

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### **2) $$ \int(2x - 5)^{4} \, dx $$**

**Solution:**

We'll use the substitution method.

1. **Let** $$ u = 2x - 5 $$.
2. **Differentiate** $$ u $$ with respect to $$ x $$:
   $$
   \frac{du}{dx} = 2 \quad \Rightarrow \quad dx = \frac{du}{2}
   $$
3. **Substitute** $$ u $$ and $$ dx $$ into the integral:
   $$
   \int (2x - 5)^4 \, dx = \int u^4 \cdot \frac{du}{2} = \frac{1}{2} \int u^4 \, du
   $$
4. **Integrate**:
   $$
   \frac{1}{2} \cdot \frac{u^5}{5} + C = \frac{u^5}{10} + C
   $$
5. **Substitute back** $$ u = 2x - 5 $$:
   $$
   \frac{(2x - 5)^5}{10} + C
   $$

**Final Answer:**
$$
\int(2x - 5)^4 \, dx = \frac{(2x - 5)^5}{10} + C
$$

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### **3) $$ \int \cos x \cdot e^{\sin x} \, dx $$**

**Solution:**

Again, we'll use substitution.

1. **Let** $$ u = \sin x $$.
2. **Differentiate** $$ u $$ with respect to $$ x $$:
   $$
   \frac{du}{dx} = \cos x \quad \Rightarrow \quad dx = \frac{du}{\cos x}
   $$
   However, since $$ du = \cos x \, dx $$, we can rearrange:
   $$
   \cos x \, dx = du
   $$
3. **Substitute** $$ u $$ and $$ du $$ into the integral:
   $$
   \int \cos x \cdot e^{\sin x} \, dx = \int e^u \, du
   $$
4. **Integrate**:
   $$
   e^u + C
   $$
5. **Substitute back** $$ u = \sin x $$:
   $$
   e^{\sin x} + C
   $$

**Final Answer:**
$$
\int \cos x \cdot e^{\sin x} \, dx = e^{\sin x} + C
$$

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**Where $$ C $$ is the constant of integration.**
$$ \int(2x - 5)^4 \, dx = \frac{(2x - 5)^5}{10} + C $$ $$ \int \cos x \cdot e^{\sin x} \, dx = e^{\sin x} + C $$

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