Q: Find $$\int \frac{\sin(x) + 1}{\cos(x)} dx.$$

$$\ln|\sec(x) + \tan(x)| + x + C$$

Q: Find $$\int \frac{1}{1 + \sin(\theta)} d\theta.$$

$$\frac{1}{2}(\tan(\frac{\theta}{2}) + \sec(\frac{\theta}{2})) + C$$

Q: Which of the following is the correct result for $$\int \frac{1}{1 + \sin(\theta)} d\theta$$ using the Weierstrass substitution $$t = \tan(\frac{\theta}{2})$$?

$$\frac{2}{1 + t^2$} + C$$ where $$t = \tan(\frac{\theta}{2})$$

Question 1

Evaluate the integral $$\int 4x^2$ + 3x \, dx.$$

Accepted Answer

$$\frac{4}{3}x^3$ + \frac{3}{2}x^2$ + C$$

Question 2

Which substitution is most appropriate for evaluating $$\int \sin(x)(1-\cos^2$(x))dx$$?

Accepted Answer

Let $$u = \cos(x)$$

Question 3

Find $$\int \frac{\sin(x) + 1}{\cos(x)} dx.$$

Accepted Answer

$$\ln|\sec(x) + 	an(x)| + x + C$$

Question 4

Evaluate $$\int \frac{x-1}{\sqrt{2-x}} dx.$$

Accepted Answer

$$-2\sqrt{2-x} + \frac{2}{3}(2-x$$)^{3/2}$$ + C$$

Question 5

Which technique is most suitable for evaluating $$\int \frac{1}{x^2$ - 2x + 10} dx$$?

Accepted Answer

Completing the square and using arctangent substitution

Question 6

Evaluate $$\int \frac{1}{x^2$ - 2x + 10} dx.$$

Accepted Answer

$$\frac{1}{3} \arctan\left(\frac{x-1}{3}\right) + C$$

Question 7

Find $$\int \frac{1}{1 + \sin(	heta)} d	heta.$$

Accepted Answer

$$\frac{1}{2}(	an(\frac{	heta}{2}) + \sec(\frac{	heta}{2})) + C$$

Question 8

A flying saucer's profile is made of a semicircle of radius $$r$$ and two identical triangles of base $$b$$ and height $$h. If$$ $$a = 10$$, $$b = 4$$, $$h = 3$$, and $$r = \frac{9}{\pi}$$, what is the total area?

Accepted Answer

$$51$$

Question 9

Which of the following integrals would be best solved using integration by parts?

Accepted Answer

$$\int x e^x dx$$

Question 10

Evaluate $$\int_{0}^{1} (t^3$ - 2)/(t + 1) dt$$ using long division and substitution.

Accepted Answer

$$\frac{1}{4} - 2\ln(2)$$

Question 11

Which of the following is a correct substitution for $$\int \sin(x)(1-\cos^2$(x))dx to $$simplify the integral?

Accepted Answer

Let $$u = \cos(x)$$, then $$du = -\sin(x)dx$$

Question 12

Which of the following real-world applications could be modeled by $$\int \frac{1}{1 + \sin(	heta)} d	heta$$?

Accepted Answer

Calculating the time for a pendulum to swing through a certain angle

Question 13

Which of the following is the correct result for $$\int \frac{1}{1 + \sin(	heta)} d	heta$$ using the Weierstrass substitution $$t = 	an(\frac{	heta}{2})$$?

Accepted Answer

$$\frac{2}{1 + t^2$} + C$$ where $$t = 	an(\frac{	heta}{2})$$

Question 14

Which of the following best describes the process of completing the square for $$x^2 - 2x + 10$?

Accepted Answer

$$x^2 - 2x + 10$$ = $$(x-1)^2 + 9$

Guided Practice: Calculus Section 8.1 – Integration Techniques

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