Calculus I · Unit 3A · exam

Unit 3A Practice Exam B

Answer key published

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Practice Exam B

Exercise

Evaluate

\int\left(3x^{-1/2}+4e^x-5\cos x\right)\,dx.

Exercise

Use four right-endpoint rectangles to estimate

\int_0^2x^2\,dx.

Exercise

Find the average value of $f(x)=x^2+1$ on $[0,3]$ .

Exercise

Differentiate

H(x)=\int_{x^2}^{2x+1}\sqrt{1+t^4}\,dt.

Exercise

Evaluate

\int\frac{x}{(1+x^2)^3}\,dx.

Exercise

Evaluate

\int x^2\ln x\,dx.

Exercise

Evaluate

\int\sin^3x\cos^2x\,dx.

Exercise

Evaluate

\int\frac{dx}{x^2+9}.

State the substitution or standard form you use.

Exercise

Use the trapezoidal rule with $n=4$ to estimate

\int_0^2x^3\,dx.

Exercise

Determine whether

\int_0^1x^{-2/3}\,dx

converges. If it converges, find its value.

Exercise

Explain why $+C$ belongs on an indefinite integral but not on a completed definite-integral evaluation.

Exercise

For each integral, name the best first method and justify the choice in one sentence:

\int x\ln x\,dx, \qquad \int\frac{2x}{x^2+7}\,dx, \qquad \int\frac{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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