Q: Which trigonometric identity is used to simplify $$\cos^2$ \theta in $$the integral $$\$$int_0$$^{\frac{\pi}{2}} \cos^2$ \theta \, d\theta$$?

$$\cos^2$ \theta = \frac{1 + \cos 2\theta}{2}$$

Q: Find the antiderivative of $$\int \frac{1}{1 + 4x^2$} \, dx.$$

$$\frac{1}{2} \tan$$^{-1}$$(2x) + C$$

Question 1

Which trigonometric identity is used to simplify $$\cos^2$ 	heta in $$the integral $$\$$int_0$$^{\frac{\pi}{2}} \cos^2$ 	heta \, d	heta$$?

Accepted Answer

$$\cos^2$ 	heta = \frac{1 + \cos 2	heta}{2}$$

Question 2

Evaluate $$\$$int_0$$^{\frac{\pi}{2}} \cos^2$ 	heta \, d	heta$$ using the appropriate trigonometric identity and substitution.

Accepted Answer

$$\frac{\pi}{4}$$

Question 3

Which substitution is most appropriate for evaluating $$\$$int_0$$^{\frac{\pi}{2}} \cos 2	heta \, d	heta$$?

Accepted Answer

$$u = 2	heta$$

Question 4

What is the value of $$\$$int_0$$^{\frac{\pi}{2}} \sin^2$ x \, dx$$?

Accepted Answer

$$\frac{\pi}{4}$$

Question 5

Which formula is used to evaluate $$\int \frac{du}{\sqrt{a^2$ - u^2$}}$$?

Accepted Answer

$$\sin$$^{-1}$$ \left( \frac{u}{|a|} \right) + C$$

Question 6

Find the antiderivative of $$\int \frac{1}{1 + 4x^2$} \, dx.$$

Accepted Answer

$$\frac{1}{2} 	an$$^{-1}$$(2x) + C$$

Question 7

Evaluate $$\int \frac{dx}{\sqrt{4 - 9x^2$}}$$ using substitution.

Accepted Answer

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

Question 8

Which of the following is the correct formula for integration by parts?

Accepted Answer

$$\int u \, dv = uv - \int v \, du$$

Question 9

Evaluate $$\int x \sin x \, dx$$ using integration by parts.

Accepted Answer

$$-x \cos x + \sin x + C$$

Question 10

When using the table method for integration by parts, what is the main advantage compared to the standard formula?

Accepted Answer

It simplifies repeated applications of integration by parts for polynomial and exponential/trigonometric products.

Question 11

Evaluate $$\int x^2$ e$$^{-2x}$$ \, dx$$ using the table method.

Accepted Answer

$$\frac{e$$^{-2x}$$}{2} \left( x^2$ + x + \frac{1}{2} \right) + C$$

Question 12

Which substitution should be used to evaluate $$\int \frac{dx}{25 + 4x^2$}$$?

Accepted Answer

$$u = 2x$$

Question 13

What is the general solution for $$\int e$$^{ax}$$ \sin(bx) \, dx$$?

Accepted Answer

$$\frac{e$$^{ax}$$}{a^2$ + b^2$} (a \sin bx - b \cos bx) + C$$

Question 14

A medical researcher models the concentration of a drug in the bloodstream as $$C(t) = e$$^{-2t}$$ \sin(3t). $$What is $$\int e$$^{-2t}$$ \sin(3t) \, dt$$?

Accepted Answer

$$\frac{e$$^{-2t}$$}{13} (-2 \sin 3t - 3 \cos 3t) + C$$

Question 15

Which of the following integrals results in an inverse tangent function?

Accepted Answer

$$\int \frac{du}{a^2$ + u^2$}$$

Question 16

A physicist needs to compute the area under the curve $$y = \frac{1}{\sqrt{9 - x^2$}}$$ from $$x = 0 to$$ $$x = 2. $$What is the value of $$\$$int_0$$^2 \frac{dx}{\sqrt{9 - x^2$}}$$?

Accepted Answer

$$\sin$$^{-1}$$ \left( \frac{2}{3} \right)$$

Question 17

Which step is necessary when using integration by parts to solve $$\int x^3$ \cos 2x \, dx$$ using the table method?

Accepted Answer

Repeatedly differentiate $$x^3$$ and integrate $$\cos 2x$$ until the polynomial vanishes.

Question 18

Evaluate $$\int (x^2$ + 2x^2$ - x + 3) \sin(3x + 4) \, dx$$ using the table method.

Accepted Answer

$$\frac{x^2$ + 2x^2$ - x + 3}{3} \cos(3x + 4) + \frac{3x^2$ + 4x - 1}{3^2$} \sin(3x + 4) + \frac{6x + 4}{3^3$} \cos(3x + 4) - \frac{6}{3^4$} \sin(3x + 4) + C$$

Advanced Techniques of Integration: Trigonometric, Inverse Trigonometric, and Exponential Integrals

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