Q: Which integral formula is correct for $$\int \tan x dx$$?

$$\int \tan x dx = \ln|\sec x| + C$$

Q: Which integral formula is correct for $$\int \sec x dx$$?

$$\int \sec x dx = \ln|\sec x + \tan x| + C$$

Question 1

Which of the following is the correct application of the Substitution Rule for the indefinite integral $$\int f(g(x))g'(x) dx$$?

Accepted Answer

$$\int f(g(x))g'(x) dx = \int f(u) du$$ where $$u = g(x)$$

Question 2

Evaluate $$\int x^3$ \cos(x^4$ + 2) dx$$ using substitution.

Accepted Answer

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

Question 3

Which integral formula is correct for $$\int 	an x dx$$?

Accepted Answer

$$\int 	an x dx = \ln|\sec x| + C$$

Question 4

Given $$f(x) is an $$even function and continuous on $$[-a, a]$$, what is $$\int_{-a}^a f(x) dx$$?

Accepted Answer

$$2 \$$int_0$$^a f(x) dx$$

Question 5

Find the area between $$y = e^x$$ and $$y = x$$ from $$x = 0 to$$ $$x = 1.$$

Accepted Answer

$$e - 1.5$$

Question 6

If $$f(x) is $$odd and continuous on $$[-a, a]$$, what is $$\int_{-a}^a f(x) dx$$?

Accepted Answer

$$0$$

Question 7

Evaluate $$\int_{0}^{1} (2x - x^2$ - x^2$) dx to $$find the area between $$y = 2x - x^2$ and y$$ = $$x^2$$ from $$x = 0 to$$ $$x = 1.$$

Accepted Answer

$$\frac{2}{3}$$

Question 8

Which substitution should be used to evaluate $$\int x^2$ \sqrt{1 + x^2$} dx$$?

Accepted Answer

$$u = 1 + x^2$

Question 9

What is the area between the curves $$y = \cos x$$ and $$y = \sin x$$ from $$x = 0 to$$ $$x = \frac{\pi}{2}$$?

Accepted Answer

$$2 - \sqrt{2}$$

Question 10

Which formula gives the area between two curves $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$\int_a^b [f(x) - g(x)] dx$$

Question 11

If $$u = g(x)$$ and $$du = g'(x) dx$$, what is the correct substitution for the definite integral $$\int_a^b f(g(x))g'(x) dx$$?

Accepted Answer

$$\int_{g(a)}^{g(b)} f(u) du$$

Question 12

Which integral formula is correct for $$\int \sec x dx$$?

Accepted Answer

$$\int \sec x dx = \ln|\sec x + 	an x| + C$$

Question 13

A region is bounded by $$y = \sqrt{x}$$ and $$y = x - 2$$ from $$x = 2 to$$ $$x = 4. $$What is the area of the blue region under $$y = \sqrt{x}$$ minus the triangle?

Accepted Answer

$$\$$int_2$$^4 \sqrt{x} dx - 2$$

Question 14

Which of the following is true for the area between curves when the bounding curves change formula within the interval?

Accepted Answer

The area is the sum of integrals for each region where the formula changes.

Question 15

For integration with respect to $$y$$, if the right boundary is $$x = f(y)$$ and the left boundary is $$x = g(y)$$, what is the formula for the area?

Accepted Answer

$$\int_c^d [f(y) - g(y)] dy$$

Question 16

Which substitution is appropriate for evaluating $$\int e$$^{5x}$$ dx$$?

Accepted Answer

$$u = 5x$$

Question 17

If $$f(x)$$ and $$g(x)$$ cross within $$[a, b]$$, how should the area between the curves be computed?

Accepted Answer

Split the region at intersection points and sum the areas where $$f(x) \geq g(x)$$ and $$g(x) \geq f(x).$$

Question 18

Which integral formula is correct for $$\int \cot x dx$$?

Accepted Answer

$$\int \cot x dx = \ln|\sin x| + C$$

Question 19

Which of the following is true for the area of a region bounded by $$y = f(x)$$ above and $$y = g(x)$$ below, from $$x = a to$$ $$x = b$$?

Accepted Answer

The area is $$\int_a^b [f(x) - g(x)] dx.$$

Indefinite Integrals, Substitution Method, and Area Between Curves

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