What is the main focus of this tutorial?

• Differentiation of trigonometric functions

• Integration by parts

• Solving differential equations

• Finding limits of functions

Which function is chosen as 'u' in the integration by parts?

• x squared

• e to the x

• sine x

• cosine x

What is the derivative of 'u' in this problem?

• cosine x

• e to the x

• sine x

• x

What is the integral of dv in the first integration by parts?

• sine x

• e to the x

• cosine x

• negative cosine x

What is the result of integrating sine x in the second integration by parts?

• cosine x

• e to the x

• negative sine x

• negative cosine x

What is the final expression for the integral of cosine x times e to the x?

• e to the x times sine x

• e to the x times cosine x

• One half e to the x times (sine x plus cosine x) plus a constant

• e to the x times (sine x minus cosine x)

Why is it necessary to perform integration by parts twice in this problem?

• To change the variable

• To eliminate the constant

• To find the derivative

• To simplify the integral

What is the purpose of defining the integral as 'i'?

• To make the equation more complex

• To simplify the algebraic manipulation

• To change the variable

• To find the derivative

What is the role of the constant term in the final solution?

• It simplifies the equation

• It changes the variable

• It represents the derivative

• It accounts for the indefinite nature of the integral

What is the key takeaway from this tutorial?

• Integration by parts can be used multiple times to solve complex integrals

• Differentiation is more important than integration

• All integrals can be solved using substitution

• Integration by parts is only useful for polynomial functions