Integration by Parts: Techniques and Examples

Integration by parts simplifies complex integrals by breaking them into manageable parts.

### Mastering Integration by Parts: Techniques and Examples
* 📘 To find the indefinite integral of \( x e^x \), we apply integration by parts, where \( u = x \) and \( dv = e^x dx \), leading to the result \( **x e^x - e^x + C** \).
* 🔍 Next, for \( x \sin x \), we again set \( u = x \) and \( dv = \sin x dx \), resulting in **\(-x \cos x + \sin x + C\)** after applying the integration by parts formula.
* 📊 When integrating \( x^2 \ln x \), we choose \( u = \ln x \) and \( dv = x^2 dx \), yielding **\(\frac{1}{3} x^3 \ln x - \frac{1}{9} x^3 + C\)** as the final answer.
* ✍ The integral of \( \ln x \) is approached by letting \( u = \ln x \) and \( dv = dx \), resulting in **\( x \ln x - x + C \)**.
* 🔄 For \( x^2 \sin x \), we utilize integration by parts twice, leading to \( **-x^2 \cos x + 2x \sin x + 2 \cos x + C** \).
* 📈 In the case of \( x^2 e^x \), we apply the integration by parts formula twice, yielding \( e^x (x^2 - 2x + 2) + C \) as the solution.
* 📏 The integral of \( \ln(x^2) \) divided by \( x \) is simplified to **\( \frac{1}{3} \ln(x^3) + C \)** through a clever application of logarithmic properties and integration by parts.
* 🎓 Lastly, for \( e^{3x} \cos(4x) \), we perform integration by parts twice, leading to the final answer of **\( \frac{3}{25} e^{3x} \cos(4x) + \frac{4}{3} e^{3x} \sin(4x) + C \)**.

Integration By Parts

This channel focuses on providing tutorial videos on organic chemistry, general chemistry, physics, algebra, trigonometry, precalculus, and calculus.

Disclaimer:  Some of the links associated with the videos in my channel may generate affiliate commissions on my behalf.  As an amazon associate, I earn from qualifying purchases that you may make through such affiliate links.


My Patreon Membership Program:  (Full Extended Videos + Worksheets)
https://www.patreon.com/MathScienceTutor


The Organic Chemistry Tutor

Integrating Functions Using Integration by Parts

Integration by parts is used to find the indefinite integral of functions like x times e^x.

By setting u = x and dv = e^x dx, we can derive the integral.

The method is also applied to other functions, such as x times sine x and x squared ln x, by strategically choosing u and dv to simplify the integration process.

The integral of the natural log of x is x ln x - x + c.

The indefinite integral of x squared sine x dx requires integration by parts twice, resulting in -x squared cosine x + 2x sine x + 2 cosine x + c.

The integral of x squared e to the x dx also uses integration by parts twice, leading to the expression x squared e to the x - 2x e to the x + 2 e to the x + c.

Finding the Anti-Derivative of x²e^x and ln(x²)

The anti-derivative of x²e^x is found using integration by parts, resulting in the final answer of e^x(x² - 2x + 2) + c.

For the integral of ln(x²)dx, integration by parts is used again, leading to the final answer of xln(x²) - 2xln(x) + 2x + c.

The power rule for logarithms is applied to rewrite ln(x²) before integrating.

The integral of 7 ln x dx is 7x ln x - 7x + c.

For the integral of e^x sin x dx, integration by parts is used twice, leading to the final answer of (1/2)e^x(sin x - cos x) + c.

The integral of ln(x^2)/x dx is approached by rewriting it as ln(x^2) * (1/x) dx.

The integral of ln(x^2)/x is solved using integration by parts, with u = ln(x^2) and dv = 1/x dx.

The solution involves rewriting the terms and combining them, leading to the equation 3 * integral(ln(x^2)/x) = ln(x)^3.

Dividing by 3 gives the final answer, along with a constant C.

The integration by parts is applied twice, first with u as sin(4x) and then with u as cos(4x).

This leads to a common integral of e^(3x)cos(4x) on both sides of the equation.

The final result is (3/25)e^(3x)(cos(4x) + (4/3)sin(4x)) + C.