Integration Techniques and Applications

The video tutorial covers the method of integration by parts, starting with an introduction to the formula and its application. It provides detailed examples, including integrating x cos x and x^2 e^(3x), demonstrating the step-by-step process. The tutorial also explains integration by substitution and offers guidelines for choosing u and dv in integration by parts. Additionally, it includes an example of integrating log x, highlighting the importance of selecting the correct functions for u and dv.

What is the formula for integration by parts?

• Integral of u dv = u - v + Integral of v du

• Integral of u dv = u + v - Integral of v du

• Integral of u dv = uv + Integral of v du

• Integral of u dv = uv - Integral of v du

In integration by parts, what do you need to identify?

• Values for u, v, and dv only

• Values for u, dv, du, and v

• Values for v and du only

• Values for u and dv only

In the example of integrating x cos x, what is chosen as u?

• 1

• sin x

• cos x

• x

What is the result of differentiating u = x in the example of x cos x?

• cos x

• x

• 1

• 0

What is the integral of dv = cos x in the example of x cos x?

• 1

• x

• cos x

• sin x

What is the final result of integrating x cos x using integration by parts?

• x sin x + x cos x + C

• x cos x + sin x + C

• x sin x - cos x + C

• x sin x + cos x + C

What method is used to integrate exponential 3x?

• Trigonometric substitution

• Partial fraction decomposition

• Integration by parts

• Integration by substitution

What is the substitution made for integrating exponential 3x?

• u = 3x

• u = x

• u = e^x

• u = 1

What is the result of integrating exponential 3x?

• (1/3)e^(x) + C

• (1/3)e^(3x) + C

• 3e^(3x) + C

• e^(3x) + C

In the example of x^2 exponential 3x, what is chosen as u?

• x^2

• exponential 3x

• 3x

• 1