Question

integral when there is a product

Answer 1

To evaluate an integral involving the product of two functions, we often use techniques such as integration by parts, substitution, or trigonometric identities. Here are a few common methods to handle integrals involving products:

### 1. **Integration by Parts**
The formula for integration by parts is:

\[
\int u \, dv = uv - \int v \, du
\]

Where:
- \(u\) is a function you differentiate,
- \(dv\) is a function you integrate.

**Example:**

Evaluate \(\int x e^{x} \, dx\).

Here, choose:
- \(u = x\) (thus \(du = dx\))
- \(dv = e^{x} \, dx\) (thus \(v = e^{x}\))

Then applying integration by parts gives:

\[
\int x e^{x} \, dx = x e^{x} - \int e^{x} \, dx = x e^{x} - e^{x} + C = e^{x}(x - 1) + C
\]

### 2. **Substitution**
If one function can be expressed in terms of another, substitution can simplify the integral.

**Example:**

Evaluate \(\int x \sin(x^2) \, dx\).

Here, use the substitution \(u = x^2\) which implies \(du = 2x \, dx\) or \(\frac{1}{2} du = x \, dx\). Thus:

\[
\int x \sin(x^2) \, dx = \frac{1}{2} \int \sin(u) \, du = -\frac{1}{2} \cos(u) + C = -\frac{1}{2} \cos(x^2) + C
\]

### 3. **Trigonometric Identities**
If the product involves trigonometric functions, known identities can be useful.

**Example:**

Evaluate \(\int \sin(x) \cos(x) \, dx\).

Using the identity \(\sin(x) \cos(x) = \frac{1}{2} \sin(2x)\), we can rewrite the integral:

\[
\int \sin(x) \cos(x) \, dx = \frac{1}{2} \int \sin(2x) \, dx = -\frac{1}{4} \cos(2x) + C
\]

### 4. **Product-to-Sum Formulas**
For certain products, using product-to-sum formulas can simplify the integral.

**Example:**

To integrate \( \int \sin(a) \sin(b) \, dx\), you can use the identity

\[
\sin(A) \sin(B) = \frac{1}{2} [\cos(A - B) - \cos(A + B)]
\]

This can then be integrated term by term.

### Conclusion
The method to use depends on the specific functions involved in the product. Each case may require a different technique, so it's important to analyze the functions and choose the most effective method based on their properties. If you have a specific integral in mind, feel free to provide it, and I'll help you tackle it step-by-step!