Question 1

Which of the following best describes the geometric interpretation of the definite integral $$\int_a^b f(x)\,dx$$?

Accepted Answer

It represents the area under the curve $$y = f(x)$$ from $$x = a to$$ $$x = b.$$

Question 2

Evaluate the following integral using substitution: $$\int 2x \sqrt{x^2$ + 1}\,dx.$$

Accepted Answer

$$\sqrt{x^2$ + 1} + C$$

Question 3

Which of the following is the correct substitution to solve $$\int x \cos(x^2$)\,dx$$?

Accepted Answer

Let $$u = x^2$, then du = 2x$$\,dx$$.

Question 4

Given $$\$$int_0$$^1 (3x^2$ + 2x)\,dx$$, what is the value of the definite integral?

Accepted Answer

$$2$$

Question 5

Which of the following is the correct result for $$\int \frac{1}{\sqrt{1-x^2$}}\,dx$$?

Accepted Answer

$$\arcsin(x) + C$$

Question 6

A tank is being filled with water at a rate given by $$r(t) = 4t$$ liters per minute. How much water is added between $$t = 0$$ and $$t = 5$$ minutes?

Accepted Answer

$$25$$ liters

Question 7

Which of the following integrals can be solved using the substitution $$u = 	an(x)$$?

Accepted Answer

$$\int \sec^2$(x)\,dx$$

Question 8

Evaluate $$\$$int_1$$^4 \frac{1}{2\sqrt{x}}\,dx.$$

Accepted Answer

$$1$$

Question 9

Which of the following is the correct antiderivative of $$f(x) = 3x^2$ - 2x + 1$$?

Accepted Answer

$$x^3$$ - $$x^2 + x + C$

Question 10

A medication is absorbed into the bloodstream at a rate $$r(t) = 5e$$^{-t}$$ mg/hour. What is the total amount absorbed from t = 0$ to t$$ = \infty$$?

Accepted Answer

$$5 mg$$

Question 11

Which of the following is the correct result for $$\int \frac{1}{x^2$ + 1}\,dx$$?

Accepted Answer

$$\arctan(x) + C$$

Question 12

If $$\int f(x)\,dx = F(x) + C$$, which of the following statements is true?

Accepted Answer

$$F'(x) = f(x)$$

Question 13

Which of the following is the correct substitution for $$\int \frac{x}{x^2$ + 1}\,dx$$?

Accepted Answer

Let $$u = x^2$ + 1$$, then $$du = 2x\,dx$$

Question 14

Evaluate $$\$$int_0$$^\pi \sin(x)\,dx.$$

Accepted Answer

$$2$$

Question 15

Which of the following is the correct result for $$\int \sec^2$(x)\,dx$$?

Accepted Answer

$$	an(x) + C$$

Question 16

A car's velocity is given by $$v(t) = 3t^2$ m/s. How far does the car travel from t = 0$ to t = 2$ seconds?

Accepted Answer

$$8$$ meters

Question 17

Which of the following is the correct result for $$\int \frac{1}{\sqrt{a^2$ - x^2$}}\,dx$$?

Accepted Answer

$$\arcsin\left(\frac{x}{a}\right) + C$$

Question 18

Which of the following is the correct result for $$\int \cos^2$(x)\,dx$$?

Accepted Answer

$$\frac{1}{2}x + \frac{1}{4}\sin(2x) + C$$

Question 19

Which of the following is the correct result for $$\int \frac{1}{x}\,dx$$?

Accepted Answer

$$\ln|x| + C$$

Introduction to Integrals and Basic Integration Techniques

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