Chapter 15 — Self-Assessment Quiz

Ten questions on $u$-substitution and integration by parts. Work each by hand, then expand the answer to check yourself. Each answer cites the section to revisit. Target 7/10 before moving to Chapter 16. Allow about 25 minutes.

Q1. Evaluate $\displaystyle\int 6x\,(x^2 + 4)^2 \, dx$.

Answer Let $u = x^2 + 4$, $du = 2x\,dx$, so $6x\,dx = 3\,du$. Then $\int 3u^2\,du = u^3 + C = (x^2 + 4)^3 + C$. Differentiate to check: $3(x^2+4)^2\cdot 2x = 6x(x^2+4)^2$. ✓ *(Section 15.1)*

Q2. Evaluate $\displaystyle\int \frac{x^2}{x^3 + 1} \, dx$.

Answer The numerator is $\tfrac13$ of the derivative of the denominator — the "derivative over function" pattern. Let $u = x^3 + 1$, $du = 3x^2\,dx$: $\frac{1}{3}\int \frac{du}{u} = \frac{1}{3}\ln|x^3 + 1| + C$. *(Section 15.2, the $\int g'/g = \ln|g|$ pattern)*

Q3. True or false: to evaluate $\displaystyle\int \cos(x^2)\,dx$ you may pull out a $\tfrac{1}{2x}$ to supply the missing factor, giving $\tfrac{1}{2x}\sin(x^2)+C$.

Answer **False.** You may adjust by a *constant*, never by a variable; $\tfrac{1}{2x}$ cannot leave the integral. In fact $\int\cos(x^2)\,dx$ has **no elementary antiderivative** (it is a Fresnel integral). Substitution rescues you only when the needed factor differs from what is present by a constant. *(Section 15.2, Common Pitfall)*

Q4. Evaluate the definite integral $\displaystyle\int_0^1 2x\,e^{x^2} \, dx$ by changing the limits.

Answer Let $u = x^2$, $du = 2x\,dx$. Limits: $x=0 \Rightarrow u=0$, $x=1 \Rightarrow u=1$. Then $\int_0^1 e^u\,du = [e^u]_0^1 = e - 1$. Note the limits change *with* the variable — never plug $x$-values into a $u$-antiderivative. *(Section 15.3, Approach 2)*

Q5. Using LIATE, which factor should be $u$ in $\displaystyle\int x^2 \ln x \, dx$, and why?

Answer $u = \ln x$. In LIATE, **L**ogarithmic outranks **A**lgebraic, so the logarithm gets the differentiating role (it simplifies to $\tfrac1x$) and $dv = x^2\,dx$. Choosing $u = x^2$ instead would leave $\int \tfrac{x^3}{3}\ln x$-type messes. *(Section 15.4, the LIATE heuristic)*

Q6. Evaluate $\displaystyle\int x \cos x \, dx$.

Answer LIATE: Algebraic before Trig, so $u = x$, $dv = \cos x\,dx$, giving $du = dx$, $v = \sin x$. Then $\int x\cos x\,dx = x\sin x - \int \sin x\,dx = x\sin x + \cos x + C$. Check: $\sin x + x\cos x - \sin x = x\cos x$. ✓ *(Section 15.5, Example 15.5.2)*

Q7. Evaluate $\displaystyle\int \ln x \, dx$.

Answer Write the integrand as $\ln x \cdot 1$ and take $u = \ln x$, $dv = dx$, so $du = \tfrac1x\,dx$, $v = x$: $x\ln x - \int x\cdot\tfrac1x\,dx = x\ln x - x + C$. The "invisible $1$" trick handles every inverse-function antiderivative. *(Section 15.5, Example 15.5.3)*

Q8. Evaluate $\displaystyle\int x^2 e^x \, dx$ (you may use the tabular method).

Answer Tabular: derivatives of $x^2$ are $x^2,\,2x,\,2,\,0$; integrals of $e^x$ stay $e^x$; signs $+,-,+$. Products: $x^2 e^x - 2x e^x + 2 e^x + C = (x^2 - 2x + 2)e^x + C$. Differentiate to confirm it returns $x^2 e^x$. ✓ *(Section 15.5, Example 15.5.5 and the Tabular Method)*

Q9. Evaluate $I = \displaystyle\int e^x \sin x \, dx$ using the rotating trick.

Answer Two passes of parts, keeping $u = e^x$ both times, regenerate $I$: you reach $I = -e^x\cos x + e^x\sin x - I$. Solving, $2I = e^x(\sin x - \cos x)$, so $I = \tfrac12 e^x(\sin x - \cos x) + C$. If you switch direction on the second pass you only get the useless identity $I = I$. *(Section 15.6, Example 15.6.1)*

Q10. You face $\displaystyle\int \frac{1}{\sqrt{4 - x^2}} \, dx$. Which technique from this chapter applies, and if none, what should you do?

Answer **Neither** $u$-substitution nor integration by parts fits: there is no inner-derivative present and no product of different types. The radical $\sqrt{a^2 - x^2}$ signals **trigonometric substitution** $x = 2\sin\theta$, which is a **Chapter 16** technique — outside this chapter's toolkit. Recognizing that an integral is *not* a substitution or parts problem is itself part of the strategy of Section 15.7. *(Sections 15.7; trig substitution deferred to Chapter 16)*

Scoring Guide

Score Interpretation
9–10 Mastery. You recognize patterns fluently and choose techniques correctly. Proceed to Chapter 16.
7–8 Solid. Review any missed item's cited section, then continue.
5–6 Developing. Re-read Sections 15.1–15.6 and redo Parts A–D of the exercises before advancing.
0–4 Revisit the chapter from Section 15.1. The skill is pure pattern recognition (Section 15.7) — it comes only from doing many problems.

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