Integration by Parts: Solving e^x sin(x) dx

This video tutorial explains how to find the integral of e^x * sin(x) using the integration by parts method. The instructor demonstrates the process step-by-step, applying the integration by parts formula twice to solve the integral. The tutorial concludes by simplifying the expression and providing the final solution, including the constant of integration.

What is the method used to solve the integral of e^x sin(x) dx?

• Integration by parts

• Partial fraction decomposition

• Trigonometric substitution

• Integration by substitution

In the first integration by parts, what is chosen as u?

• sin(x)

• x

• cos(x)

• e^x

What is the integral of sin(x) dx?

• e^x

• -e^x

• cos(x)

• -cos(x)

After applying the integration by parts formula, what is the expression for u * v?

• e^x * sin(x)

• -e^x * cos(x)

• e^x * cos(x)

• -e^x * sin(x)

What is the next step after obtaining the expression -e^x cos(x) + ∫e^x cos(x) dx?

• Differentiate the expression

• Use trigonometric identities

• Use substitution

• Apply integration by parts again

In the second integration by parts, what is chosen as dv?

• x dx

• sin(x) dx

• cos(x) dx

• e^x dx

What is the integral of cos(x) dx?

• -sin(x)

• sin(x)

• -e^x

• e^x

What is the expression obtained after the second integration by parts?

• e^x cos(x) - ∫e^x cos(x) dx

• -e^x cos(x) + ∫e^x cos(x) dx

• e^x sin(x) - ∫e^x sin(x) dx

• -e^x sin(x) + ∫e^x sin(x) dx

What is the final step to solve the integral of e^x sin(x) dx?

• Differentiate the result

• Multiply by 1/2

• Use a trigonometric identity

• Add a constant C

What is the final solution for the integral of e^x sin(x) dx?

• e^x sin(x) - e^x cos(x) + C

• 1/2 (e^x sin(x) - e^x cos(x)) + C

• 1/2 (e^x cos(x) - e^x sin(x)) + C

• e^x sin(x) + e^x cos(x) + C