$$\frac{1}{2} \ln(x^2$+3) - \sqrt{3} \tan$$^{-1}$$\left(\frac{x}{\sqrt{3}}\right) + C$$

Let $$x = \sqrt{3} \tan \theta$$

$$\sec \theta = \frac{x^2$+3}{\sqrt{3}}$$

$$\theta = \tan$$^{-1}$$\left(\frac{x}{\sqrt{3}}\right)$$

Rewrite $$\tan^2$ \theta as$$ $$\sec^2$ \theta - 1$$

Question 1

Evaluate the integral $$\int \frac{x-1}{x^2$+3} \, dx$$ using appropriate substitution and trigonometric identities.

Accepted Answer

$$\frac{1}{2} \ln(x^2$+3) - \sqrt{3} 	an$$^{-1}$$\left(\frac{x}{\sqrt{3}}ight) + C$$

Question 2

Which substitution is most appropriate for evaluating $$\int \frac{x}{x^2$+3} \, dx$$?

Accepted Answer

Let $$u = x^2$ + 3$$

Question 3

Given $$\int \frac{1}{x^2$+3} \, dx$$, which trigonometric substitution is used in the solution?

Accepted Answer

Let $$x = \sqrt{3} 	an 	heta$$

Question 4

What is the value of $$\int 	an^2$ 	heta \, d	heta$$?

Accepted Answer

$$	an 	heta - 	heta + C$$

Question 5

Evaluate $$\int x e$$^{2x}$$ \, dx$$ using integration by parts.

Accepted Answer

$$\frac{1}{2} x e$$^{2x}$$ - \frac{1}{4} e$$^{2x}$$ + C$$

Question 6

For the integral $$\int \sin^2$(2t) \, dt$$, which substitution is used first?

Accepted Answer

Let $$u = 2t$$

Question 7

After substituting $$u = 2t in$$ $$\int \sin^2$(2t) \, dt$$, what is the next step?

Accepted Answer

Integrate by parts with $$w = \cos(u)$$

Question 8

What is the final answer for $$\int \sin^2$(2t) \, dt as $$shown in the notes?

Accepted Answer

$$-\frac{1}{2} \cos(2t) + \frac{1}{6} \cos^3$(2t) + C$$

Question 9

In the process of integrating $$\int \frac{1}{x^2$+3} \, dx$$, what is the value of $$\sec 	heta in $$terms of $$x$$?

Accepted Answer

$$\sec 	heta = \frac{x^2$+3}{\sqrt{3}}$$

Question 10

Which of the following is the correct result for $$\int \frac{x}{x^2$+3} \, dx$$?

Accepted Answer

$$\frac{1}{2} \ln(x^2$+3) + C$$

Question 11

If $$	an 	heta = \frac{x}{\sqrt{3}}$$, what is $$	heta in $$terms of $$x$$?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{x}{\sqrt{3}}ight)$$

Question 12

Which integration technique is used to solve $$\int x e$$^{2x}$$ \, dx$$?

Accepted Answer

Integration by parts

Question 13

What is the integral $$\int \sec^2$ 	heta \, d	heta$$?

Accepted Answer

$$	an 	heta + C$$

Question 14

Which of the following best describes the use of trigonometric identities in evaluating $$\int 	an^2$ 	heta \, d	heta$$?

Accepted Answer

Rewrite $$	an^2$ 	heta as$$ $$\sec^2$ 	heta - 1$$

Question 15

In the context of integration by parts, if $$u = x$$ and $$dv = e$$^{2x}$$ dx$$, what is $$du$$?

Accepted Answer

$$dx$$

Question 16

Which real-world application could use the integral $$\int x e$$^{2x}$$ \, dx$$?

Accepted Answer

Modeling exponential growth with a linear rate, such as population growth with a time-dependent rate

Integration Practice: Techniques and Applications

Study Guide - Practice Questions