Question 1

Find the integral using an appropriate trigonometric substitution.

Accepted Answer

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Question 2

Use long division to evaluate the integral.  The choices are rounded to 3 decimal places.

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Question 3

Evaluate the integral.

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...

Question 4

The region under the graph of  on the interval [1, 2] is revolved about the x-axis. Find the volume of the resulting solid.

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Question 5

Eight milligrams of a dye are injected into a vein leading the an individual's heart. The concentration of dye in the aorta (in milligrams per liter) measured at 2-sec intervals is shown in the accompanying table. Use Simpson's Rule with  and the formula  to estimate the person's cardiac output, where D is the quantity of dye injected in milligrams,  is the concentration of the dye in the aorta, and R is measured in liters per minute. Round your answer to one decimal place.

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Question 6

Evaluate the integral using the indicated trigonometric substitution.

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Question 7

Find the integral using an appropriate trigonometric substitution.

Accepted Answer

The integral can be solved by using the trigonometric substitution $x=\sqrt{a^2-u^2}$, which gives $dx=-\frac{u}{\sqrt{a^2-u^2}}du$ and $\sqrt{a^2-x^2}=u$. Plugging these into the integral, we get:

$$\int{\frac{dx}{x^2\sqrt{a^2-x^2}}}=\int{\frac{-1/a^2}{(1-(x/a)^2)^{3/2}}dx}$$

Now, let $x/a=\sin	heta$, so $dx=a\cos	heta d	heta$. Substituting this in and simplifying:

$$\int{\frac{-a\cos	heta d	heta}{a^2\sin^2	heta\sqrt{1-\sin^2	heta}}}=-\int{\frac{\cos	heta d	heta}{a\sin^2	heta}}$$

Using the identity $\sin^2	heta=1-\cos^2	heta$, this becomes:

$$-\int{\frac{d(\sin	heta)}{a\sin^2	heta}}=\frac{1}{a\sin	heta}+C=\frac{1}{a\sqrt{1-(x/a)^2}}+C$$

Therefore, the answer is (B).

Question 8

Evaluate the integral using an appropriate trigonometric substitution.

Accepted Answer

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Question 9

Evaluate the integral.

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Question 10

Find the integral using an appropriate trigonometric substitution.

Accepted Answer

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Question 11

Evaluate the integral using the indicated trigonometric substitution.

Accepted Answer

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Question 12

A corporation is building a complex of homes, offices, stores, schools, and churches in a rural community. As a result of this development, the planners have estimated that the community's population (in thousands) t years from now will be given by  .
What will the average population of the community be over the next 10 years?

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Question 13

Find the integral.

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Question 14

Make a substitution to express the integrand as a rational function and then evaluate the integral.  Round the answer to four decimal places.

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Question 15

Find the integral.

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Question 16

Evaluate the integral.

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...

Question 17

Find the integral using an appropriate trigonometric substitution.

Accepted Answer

The integral involves the square root of the quadratic expression $4x^2+5$. This suggests that we can use a trigonometric substitution $x=\frac{\sqrt{5}}{2}	an	heta$. Then, we have 
\begin{align*}
4x^2+5 &= 4\left(\frac{\sqrt{5}}{2}	an	hetaight)^2+5 \
&= 5\sec^2	heta,
\end{align*}
and $dx = \frac{\sqrt{5}}{2}\sec^2	heta d	heta$. Substituting these expressions into the given integral, we get 
\begin{align*}
\int\frac{\sqrt{4x^2+5}}{x}dx &= \int \frac{\sqrt{5\sec^2	heta}}{\frac{\sqrt{5}}{2}	an	heta}\cdot\frac{\sqrt{5}}{2}\sec^2	heta d	heta \
&= \int \frac{5}{2}\cdot\frac{\sec^3	heta}{	an	heta}d	heta\
&= \int \frac{5}{2}\sec^2	heta d	heta \
&= \frac{5}{2}	an	heta + C \
&= \frac{5}{2}	an\left(\arctan\frac{2x}{\sqrt{5}}ight) + C \
&= \frac{5}{2}\cdot\frac{2x}{\sqrt{5}}+C \
&= x\sqrt{5}+C.
\end{align*}
Hence, the correct choice is $\boxed{	ext{(B)}}$.

Question 18

Find the integral.

Accepted Answer

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Question 19

Find the integral.

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...

Question 20

Find the volume of the resulting solid if the region under the curve  from  to  is rotated about the x-axis. Round your answer to four decimal places.

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The answer of Find the volume of the resulting solid...

Quiz 7: Techniques of Integration

Find the integral using an appropriate trigonometric substitution.

Use long division to evaluate the integral. The choices are rounded to 3 decimal places.

Evaluate the integral.

The region under the graph of on the interval [1, 2] is revolved about the x-axis. Find the volume of the resulting solid.

Evaluate the integral using the indicated trigonometric substitution.

Find the integral using an appropriate trigonometric substitution.

Evaluate the integral using an appropriate trigonometric substitution.

Evaluate the integral.

Find the integral using an appropriate trigonometric substitution.

Evaluate the integral using the indicated trigonometric substitution.

Find the integral.

Make a substitution to express the integrand as a rational function and then evaluate the integral. Round the answer to four decimal places.

Find the integral.

Evaluate the integral.

Find the integral using an appropriate trigonometric substitution.

Find the integral.

Find the integral.

Find the volume of the resulting solid if the region under the curve from to is rotated about the x-axis. Round your answer to four decimal places.