Calculus I · Unit 3A · answer key

Unit 3A Practice Exam B Answer Key

Finish an honest attempt first. Then compare one numbered response at a time, locate the first line where your reasoning diverged, and retry without the key open.

Answer 1

Problem 1

6\sqrt{x}+4e^x-5\sin x+C.

Answer 2

Problem 2

$\Delta x=\frac12$ , with right endpoints $\frac12,1,\frac32,2$ :

R_4=\frac12\left(\frac14+1+\frac94+4\right)=\frac{15}{4}.

Answer 3

Problem 3

f_{\mathrm{avg}}=\frac13\int_0^3(x^2+1)dx =\frac13\left[\frac{x^3}{3}+x\right]_0^3=4.

Answer 4

Problem 4

Apply FTC to both bounds:

H'(x)=2\sqrt{1+(2x+1)^4}-2x\sqrt{1+x^8}.

Answer 5

Problem 5

Let $u=1+x^2$ :

\int\frac{x}{(1+x^2)^3}dx =\frac12\int u^{-3}du =-\frac{1}{4(1+x^2)^2}+C.

Answer 6

Problem 6

Integration by parts gives

\int x^2\ln x\,dx =\frac{x^3}{3}\ln x-\frac{x^3}{9}+C.

Answer 7

Problem 7

Write $\sin^3x=(1-\cos^2x)\sin x$ and use $u=\cos x$ :

-\frac{\cos^3x}{3}+\frac{\cos^5x}{5}+C.

Answer 8

Problem 8

Using $x=3\tan\theta$ , or the standard arctangent form,

\int\frac{dx}{x^2+9}=\frac13\arctan\left(\frac{x}{3}\right)+C.

Answer 9

Problem 9

With $h=\frac12$ ,

T_4=\frac{h}{2}\left[f(0)+2f(0.5)+2f(1)+2f(1.5)+f(2)\right] =\frac{17}{4}.

Answer 10

Problem 10

\int_0^1x^{-2/3}dx =\lim_{a\to0^+}\left[3x^{1/3}\right]_a^1=3.

The integral converges.

Answer 11

Problem 11

Indefinite integration reverses differentiation and therefore determines a family differing by constants. Definite integration subtracts endpoint values, so any antiderivative constant cancels.

Answer 12

Problem 12

Use integration by parts for $\int x\ln x\,dx$ ; substitution $u=x^2+7$ for $\int 2x/(x^2+7)dx$ ; and partial fractions after factoring $x^2-4$ for $\int dx/(x^2-4)$ .

After the explanation

Use the section idea

Reading lens

Mixed integral work tests recognition: classify the output and structure before committing to a method.

Mental model

A complete response carries setup, method, computation, bounds or constants, units, interpretation, and an independent verification.

Decision

Attempt the entire problem first, then use one answer at a time to locate the earliest reasoning decision that needs repair.

Common trap

Reading a key before modeling the problem or treating every mismatch as algebra when the first error was conceptual.

Check yourself

Can you reproduce the reasoning without the key and explain why the final form fits the question?

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