Integration by parts

Here are the key points for studying "Integration by Parts":

1. Formula: Understand the integration by parts formula:

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

   where $u$ and $dv$ are chosen parts of the integrand.

2. Choosing $u$ and $dv$: Learn how to select $u$ and $dv$ effectively. A common strategy is to choose $u$ as a function that simplifies upon differentiation and $dv$ as a function that is easy to integrate.

3. Differentiation and Integration: Familiarize yourself with how to differentiate $u$ to find $du$ and integrate $dv$ to find $v$.

4. Iterative Process: Recognize that integration by parts may require multiple applications, especially when the resulting integral is still complex.

5. Practice Problems: Work on a variety of problems to gain proficiency in applying the method, including polynomial, logarithmic, and trigonometric integrands.

6. Examples: Learn from examples that illustrate the process, including scenarios where integration by parts simplifies the calculation significantly.

7. Special Cases: Be aware of special cases where integration by parts can be used to derive standard integral formulas or solve specific types of integrals efficiently.

Remember to practice regularly to solidify these concepts and improve your integration skills.

When studying "Integration by parts" using the example ∫x⋅cos(x)dx, focus on the following key points:

1. Integration by Parts Formula: The formula is ∫u dv = uv - ∫v du, where you need to identify parts of the integrand as u and dv.

2. Choosing u and dv:

   • Select u as a function that simplifies upon differentiation (in this case, u = x).
   • Let dv be the remaining portion of the integrand (here, dv = cos(x)dx).

3. Compute du and v:

   • Differentiate u to find du (du = dx).
   • Integrate dv to find v (v = ∫cos(x)dx = sin(x)).

4. Substitute into the Formula: Plug u, du, v, and dv into the integration by parts formula.

5. Evaluate the Integral: Simplify and evaluate the remaining integral ∫sin(x)dx.

6. Combine Results: After integrating, combine the results from your substitution to find the final answer.

7. Don't Forget the Constant of Integration: Always add the constant C at the end of your result.

By focusing on these key steps, you can effectively apply the integration by parts technique to similar problems.

Key Points for "Integration by Parts: ∫ln(x)dx"

1. Integration by Parts Formula:

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

2. Choosing $u$ and $dv$:

   • Set $u = \ln(x)$ (because its derivative simplifies the integral).
   • Then, $du = \frac{1}{x} \, dx$.
   • Choose $dv = dx$ (which is straightforward to integrate).

3. Finding $v$:

   • Integrate $dv$: $v = x$.

4. Apply Integration by Parts:

   $$\int \ln(x) \, dx = x \ln(x) - \int x \cdot \frac{1}{x} \, dx$$

   Simplifying gives:

   $$\int \ln(x) \, dx = x \ln(x) - \int 1 \, dx$$

5. Integrate the Remaining Term:

   $$\int 1 \, dx = x$$

6. Combine Results:

   $$\int \ln(x) \, dx = x \ln(x) - x + C$$

   where $C$ is the constant of integration.

Main Result:

When studying "Integration by Parts" for the integral ∫x²⋅eˣ dx, focus on the following key points:

1. Integration by Parts Formula:

   • The formula is derived from the product rule of differentiation:
     $$\int u \, dv = uv - \int v \, du$$
   • Choose $u$ and $dv$ strategically to simplify the integral.

2. Choosing $u$ and $dv$:

   • Typically, $u$ is chosen as the polynomial part (x²) since its derivative will simplify the integral.
   • $dv$ is chosen as the remaining part (eˣ dx).

3. Differentiation and Integration:

   • Differentiate $u$ to find $du$ and integrate $dv$ to find $v$:
     • $u = x²$ → $du = 2x \, dx$
     • $dv = eˣ dx$ → $v = eˣ$

4. Applying the Formula:

   • Substitute into the integration by parts formula:
     $$\int x² eˣ \, dx = x² eˣ - \int eˣ (2x) \, dx$$

5. Second Integration by Parts:

   • Now, you may need to apply integration by parts again on the integral $\int 2x eˣ \, dx$ using the same strategy.
   • Choose $u = 2x$ and $dv = eˣ dx$.

6. Final Combination:

   • After solving the second integral, combine all parts to get the final result, maintaining any necessary constants of integration.

7. Check Your Work:

   • Differentiate your final result to confirm it matches the original integrand x² eˣ.

By mastering these points, you will develop a solid understanding of how to use integration by parts effectively, especially with integrals involving products of polynomials and exponential functions.

To study the integration of the function $\int e^x \cos(x) \, dx$ using integration by parts, focus on the following key points:

1. Integration by Parts Formula: Understand the formula:

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

   where you will need to choose $u$ and $dv$ appropriately.

2. Choosing $u$ and $dv$:

   • Set $u = e^x$ and $dv = \cos(x) \, dx$.
   • Differentiate $u$ to find $du$ and integrate $dv$ to find $v$.

3. Calculate $du$ and $v$:

   • $du = e^x \, dx$
   • $v = \sin(x)$

4. Apply the formula: Substitute into the integration by parts formula:

   $$\int e^x \cos(x) \, dx = e^x \sin(x) - \int \sin(x) e^x \, dx$$

5. Second Integral: Now, focus on $\int e^x \sin(x) \, dx$ and repeat the integration by parts.

6. Using Integration by Parts Again: Choose $u = e^x$ and $dv = \sin(x) \, dx$. Determine $du$ and $v$ again.

7. Set up the new equation: After applying integration by parts again, you get another equation that will lead you back to the original integral.

8. Solve for the original integral: Combine the results to isolate the original integral and solve for it.

9. Final Result: Write the solution in terms of the original integral, leading to a formula involving $e^x$, $\sin(x)$, and $\cos(x)$.

10. Conclusion: Use the derived equation to express $\int e^x \cos(x) \, dx$ in a closed form.

These key points will guide you through the process of solving $\int e^x \cos(x) \, dx$ effectively using integration by parts.

When studying "Integration by Parts: Definite Integrals," focus on the following key points:

1. Formula: Understand the integration by parts formula:

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

   where $u$ and $dv$ must be chosen appropriately.

2. Choosing $u$ and $dv$: Apply the LIATE rule (Logarithmic, Inverse, Algebraic, Trigonometric, Exponential) to help select $u$ and $dv$.

3. Differentiation and Integration: Compute $du$ (the derivative of $u$) and $v$ (the integral of $dv$) after making your selections.

4. Bounds Adjustment: For definite integrals, evaluate the expression $uv$ at the bounds, and subtract the integral evaluated at the same bounds:

   $$\int_a^b u \, dv = [uv]_a^b - \int_a^b v \, du$$

5. Iterative Application: Sometimes, integration by parts may need to be applied more than once if the resulting integral is still complex.

6. Simplifying the Integral: Focus on simplifying the integral $\int v \, du$. If it becomes simpler, take advantage of this to obtain the final result.

7. Examples Practice: Work through several examples to reinforce understanding and become adept at selecting $u$ and $dv$.

8. Recognize Patterns: After more practice, recognize integral patterns that can simplify the process, especially with common functions.

By focusing on these key elements, you can effectively master the application of integration by parts to definite integrals.