The integration by parts formula is a powerful technique for solving integrals where the standard methods (like basic integration rules or substitution) are difficult to apply. It comes from the product rule of differentiation and is useful when the integrand is a product of two functions.

The formula for integration by parts is:

$\int u \, dv = uv - \int v \, du$

Where:

• $u$ is a function that you choose to differentiate.
• $dv$ is a function that you choose to integrate.

Steps to apply integration by parts:

1. Identify parts of the integrand as $u$ and $dv$.
2. Differentiate $u$ to get $du$.
3. Integrate $dv$ to get $v$.
4. Substitute into the formula: $\int u \, dv = uv - \int v \, du$.
5. Solve the resulting integral.

Example:

For the integral $\int x e^x \, dx$:

• Let $u = x$ and $dv = e^x \, dx$.
• Differentiate $u$: $du = dx$.
• Integrate $dv$: $v = e^x$.

Now apply the formula:

$\int x e^x \, dx = x e^x - \int e^x \, dx$

The remaining integral is easy to solve:

$\int e^x \, dx = e^x$

So the final solution is:

$x e^x - e^x + C$

Where $C$ is the constant of integration.

Would you like more details on how to apply this method or see another example?

5 related questions:

1. How do you choose $u$ and $dv$ in integration by parts?
2. Can integration by parts be used multiple times for complex integrals?
3. What are some common integrals that can be solved using integration by parts?
4. How does the formula for integration by parts derive from the product rule?
5. What happens if the resulting integral after applying parts is the same as the original?

Tip:

When choosing $u$ in integration by parts, try the LIATE rule (Logarithmic, Inverse trig, Algebraic, Trig, Exponential) to prioritize which function to differentiate.