Integration by parts is most useful when dealing with:

a) Exponential functions

b) Trigonometric functions

c) Polynomial functions

d) Rational functions

Which of the following is true regarding definite integrals?

a) They can be negative.

b) They always evaluate to zero.

c) They represent the area under the curve.

d) They only apply to continuous functions.

The definite integral of \(e^x \) from 0 to 1 is equal to:

a) \(e^{-1}\)

b) \(e + 1\)

c) \(e^2 – 1\)

d) \( e^2 + 1\)

If \(F(x) = \int_{0}^{x} f(t) dt\), then according to the Fundamental Theorem of Calculus,  F'(x)  is equal to:

a) f(x)

b) F(x)

c) \(\int_{0}^{x} f'(t) dt\)

d) \(\frac{d}{dx} \int_{0}^{x} f(t) dt\)

Which technique is most appropriate for evaluating integrals involving rational functions?

a) Substitution

b) Partial fractions

c) Integration by parts

d) Trigonometric substitution

The integral \( \int \frac{1}{1+x^2} dx\) can be evaluated using:

a) Trigonometric substitution

b) Partial fractions

c) Integration by parts

d) Substitution

Which method is typically used to evaluate integrals of products of functions?

a) Substitution

b) Partial fractions

c) Integration by parts

d) Trigonometric substitution

Which of the following statements best describes integration as the inverse process of differentiation?

a) Integration finds the derivative of a function.

b) Integration finds the area under a curve.

c) Integration finds the original function given its derivative.

d) Integration finds the slope of a tangent line.

The integral of \( \sin(x) \cos(x)\) with respect to x is:

a) \(\frac{1}{2}\sin^2(x) + C \)

b) \(-\frac{1}{2}\cos^2(x) + C\)

c) \( -\frac{1}{2}\sin^2(x) + C\)

d) \(\frac{1}{2}\cos^2(x) + C\)

The integral \(\int \frac{x^2 + 1}{x} dx\) can be evaluated using:

a) Trigonometric substitution

b) Partial fractions

c) Integration by parts

d) Substitution