Chapter 15 — Exercises

These exercises drill the two techniques of Chapter 15 — $u$-substitution (Sections 15.1–15.3) and integration by parts (Sections 15.4–15.6) — plus the strategy of deciding which to use (Section 15.7) and combining them (Section 15.8). Everything here stays inside those two methods; trigonometric substitution and partial fractions wait for Chapter 16, and improper integrals for Chapter 17.

Work each problem by hand, and verify every indefinite answer by differentiating it — that single differentiation, as Section 15.1 stresses, catches nearly every error. Use sympy (Section 15.9) only to confirm after you have committed to an answer.

Difficulty tiers:

Tier counts: ⭐ (10) · ⭐⭐ (12) · ⭐⭐⭐ (10) · ⭐⭐⭐⭐ (6) — 38 problems total.

Part A — Basic $u$-Substitution (⭐)

Reverse the chain rule. In each case the inner function and (a constant times) its derivative are both present. (Section 15.1)

A1. ⭐ Evaluate $\displaystyle\int 2x\,(x^2 + 7)^4 \, dx$.

A2. ⭐ Evaluate $\displaystyle\int \cos(5x) \, dx$.

A3. ⭐ Evaluate $\displaystyle\int e^{4x} \, dx$.

A4. ⭐ Evaluate $\displaystyle\int 3x^2 \sin(x^3) \, dx$.

A5. ⭐ Evaluate $\displaystyle\int \frac{1}{5x + 2} \, dx$.

A6. ⭐ Evaluate $\displaystyle\int (2x + 1)\,(x^2 + x)^9 \, dx$.

A7. ⭐ Evaluate $\displaystyle\int \frac{2x}{x^2 + 9} \, dx$.

A8. ⭐ Evaluate $\displaystyle\int \sin^4 x \cos x \, dx$. (Section 15.2, Example 15.2.4)

A9. ⭐ Evaluate $\displaystyle\int e^{\sin x} \cos x \, dx$.

A10. ⭐ Evaluate $\displaystyle\int \frac{(\ln x)^2}{x} \, dx$. (Section 15.2, Example 15.2.5)

Part B — $u$-Substitution Requiring a Choice or Constant Adjustment (⭐⭐)

Here the needed derivative is present only up to a constant, or you must express a leftover $x$ in terms of $u$. (Sections 15.1–15.2)

B1. ⭐⭐ Evaluate $\displaystyle\int x \, e^{x^2} \, dx$.

B2. ⭐⭐ Evaluate $\displaystyle\int x^2 \sqrt{x^3 + 1} \, dx$.

B3. ⭐⭐ Evaluate $\displaystyle\int \frac{x}{\sqrt{1 - x^2}} \, dx$.

B4. ⭐⭐ Evaluate $\displaystyle\int \cot x \, dx$. (Hint: write $\cot x = \dfrac{\cos x}{\sin x}$ and use the derivative-over-function pattern, Section 15.2.)

B5. ⭐⭐ Evaluate $\displaystyle\int x \sqrt{x - 3} \, dx$. (Leftover $x$: solve $x = u + 3$, Section 15.1.)

B6. ⭐⭐ Evaluate $\displaystyle\int \frac{x^2}{(x^3 + 5)^2} \, dx$.

B7. ⭐⭐ Evaluate $\displaystyle\int \frac{e^{1/x}}{x^2} \, dx$.

B8. ⭐⭐ Evaluate $\displaystyle\int \frac{x}{\sqrt{x + 4}} \, dx$.

B9. ⭐⭐ Evaluate $\displaystyle\int \frac{\sec^2 x}{\sqrt{\tan x}} \, dx$.

B10. ⭐⭐ Evaluate $\displaystyle\int \frac{1}{x \ln x} \, dx$.

B11. ⭐⭐ Evaluate $\displaystyle\int \frac{6x - 1}{3x^2 - x + 4} \, dx$.

B12. ⭐⭐ Evaluate $\displaystyle\int x^5 \sqrt{x^2 + 1} \, dx$. (Leftover: $x^4 = (u-1)^2$, compare Section 15.2, Example 15.2.7.)

Part C — Definite Integrals by Substitution (⭐⭐ to ⭐⭐⭐)

Use Approach 2 from Section 15.3: change the variable and the limits together. Never evaluate a $u$-antiderivative at the original $x$-limits.

C1. ⭐⭐ Evaluate $\displaystyle\int_0^2 2x\,(x^2 + 1)^3 \, dx$.

C2. ⭐⭐ Evaluate $\displaystyle\int_0^{\pi/2} \cos x \, e^{\sin x} \, dx$.

C3. ⭐⭐ Evaluate $\displaystyle\int_1^4 \frac{1}{\sqrt{x}\,(1 + \sqrt{x})} \, dx$. (Let $u = 1 + \sqrt{x}$.)

C4. ⭐⭐⭐ Evaluate $\displaystyle\int_1^e \frac{(\ln x)^3}{x} \, dx$. (Compare Example 15.3.3.)

C5. ⭐⭐⭐ Evaluate $\displaystyle\int_0^1 \frac{x}{x^2 + 1} \, dx$ and state the answer as a single logarithm.

C6. ⭐⭐⭐ Evaluate $\displaystyle\int_{-2}^{2} \frac{x^3}{x^4 + 1} \, dx$ by recognizing parity before computing. (Section 15.3, Example 15.3.2.)

Part D — Integration by Parts (⭐⭐ to ⭐⭐⭐)

Apply $\displaystyle\int u\,dv = uv - \int v\,du$, choosing $u$ by LIATE (Section 15.4). Verify by differentiating.

D1. ⭐⭐ Evaluate $\displaystyle\int x \, e^{2x} \, dx$.

D2. ⭐⭐ Evaluate $\displaystyle\int x \sin(3x) \, dx$.

D3. ⭐⭐ Evaluate $\displaystyle\int \ln(2x) \, dx$. (Compare Example 15.5.3.)

D4. ⭐⭐ Evaluate $\displaystyle\int x \ln x \, dx$. (Which factor is $u$? LIATE: Logarithmic beats Algebraic.)

D5. ⭐⭐ Evaluate $\displaystyle\int \arcsin x \, dx$. (Inverse-trig standing alone; the leftover is a $u$-substitution.)

D6. ⭐⭐⭐ Evaluate $\displaystyle\int x^2 \ln x \, dx$.

D7. ⭐⭐⭐ Evaluate $\displaystyle\int x^2 e^{-x} \, dx$ using the tabular method (Section 15.5).

D8. ⭐⭐⭐ Evaluate $\displaystyle\int x^3 \cos x \, dx$ using the tabular method.

D9. ⭐⭐⭐ Evaluate $\displaystyle\int_0^1 x \, e^{-x} \, dx$ (definite integration by parts).

D10. ⭐⭐⭐ Evaluate $\displaystyle\int \arctan(2x) \, dx$. (Parts feeds a substitution, as in Example 15.5.4.)

Part E — The Rotating Trick (⭐⭐⭐)

Two passes of parts regenerate the original integral; solve for it algebraically. (Section 15.6)

E1. ⭐⭐⭐ Evaluate $\displaystyle\int e^{2x} \cos x \, dx$.

E2. ⭐⭐⭐ Evaluate $\displaystyle\int e^{-x} \sin(2x) \, dx$.

Part F — Combined Techniques and Applications (⭐⭐⭐⭐)

These require sequencing substitution and parts, or applying the chapter's tools to a model. At least two different applied fields appear. (Sections 15.7–15.8, 15.10)

F1. ⭐⭐⭐⭐ Evaluate $\displaystyle\int \sin(\sqrt{x}) \, dx$ by substituting first, then integrating by parts. (Compare Example 15.8.1.)

F2. ⭐⭐⭐⭐ Evaluate $\displaystyle\int x^5 e^{-x^2} \, dx$. (Substitute $u = x^2$, then by parts twice — a higher Gaussian moment than Example 15.8.3.)

F3. ⭐⭐⭐⭐ (Statistics.) For the exponential density $f(x) = \lambda e^{-\lambda x}$ on $[0,\infty)$, the second moment is $E[X^2] = \displaystyle\int_0^\infty x^2 \lambda e^{-\lambda x}\,dx$. Compute it by parts (twice), then combine with $E[X] = 1/\lambda$ (Section 15.10) to find the variance $\operatorname{Var}(X) = E[X^2] - (E[X])^2$.

F4. ⭐⭐⭐⭐ (Physics.) A particle moves under a position-dependent force $F(x) = x\,e^{-x}$ (newtons, $x$ in meters). The work done from $x = 0$ to $x = 2$ is $W = \displaystyle\int_0^2 x e^{-x}\,dx$. Compute $W$, then explain in one sentence why the work is less than the rough estimate $F(1)\cdot 2 = 2e^{-1}$.

F5. ⭐⭐⭐⭐ (Economics.) A market has demand $D(p) = (100 - 5p)\,e^{-0.1 p}$ for $0 \le p \le 20$. Find the total "willingness-to-pay area" $\displaystyle\int_0^{20} D(p)\,dp$ — a product of a linear and an exponential factor that needs one integration by parts. Leave the answer in exact form, then give a two-decimal numerical value.

F6. ⭐⭐⭐⭐ (Strategy, Section 15.7.) For each integrand, state which technique you would try first and why — do not compute: (a) $\displaystyle\int \frac{x^2}{x^3 + 1}\,dx$ (b) $\displaystyle\int x^2 \ln x\,dx$ (c) $\displaystyle\int x\cos(x^2)\,dx$ (d) $\displaystyle\int e^x \cos x\,dx$ (e) $\displaystyle\int \frac{1}{\sqrt{9 - x^2}}\,dx$ (f) $\displaystyle\int \frac{2x+3}{x^2 + 3x + 5}\,dx$.

Answers and Hints

Full worked solutions to the odd-numbered problems in each Part appear in Appendix: Answers to Selected Exercises. Spot-check answers and the key step for representative problems are below. Always confirm by differentiating.

Part A.

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A1. $u = x^2 + 7$: $\frac{1}{5}(x^2+7)^5 + C$. A2. $\frac{1}{5}\sin(5x) + C$. A3. $\frac{1}{4}e^{4x} + C$. A4. $u = x^3$: $-\cos(x^3) + C$. A5. $u = 5x + 2$: $\frac{1}{5}\ln|5x+2| + C$. A6. $u = x^2 + x$: $\frac{1}{10}(x^2+x)^{10} + C$. A7. $u = x^2 + 9$: $\ln(x^2+9) + C$. A8. $u = \sin x$: $\frac{1}{5}\sin^5 x + C$. A9. $u = \sin x$: $e^{\sin x} + C$. A10. $u = \ln x$: $\frac{1}{3}(\ln x)^3 + C$.

Part B (selected).

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B1. $u = x^2$, $x\,dx = \tfrac12 du$: $\frac{1}{2}e^{x^2} + C$. B3. $u = 1 - x^2$, $x\,dx = -\tfrac12 du$: $-\sqrt{1 - x^2} + C$. B5. $u = x - 3$, $x = u + 3$: $\int (u+3)u^{1/2}\,du = \frac{2}{5}(x-3)^{5/2} + 2(x-3)^{3/2} + C$. B7. $u = 1/x$, $du = -\tfrac{1}{x^2}dx$: $-e^{1/x} + C$. B9. $u = \tan x$, $du = \sec^2 x\,dx$: $2\sqrt{\tan x} + C$. B11. Numerator $6x - 1$ is exactly the derivative of $3x^2 - x + 4$: $\ln|3x^2 - x + 4| + C$.

Part C (selected).

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C1. $u = x^2 + 1$, limits $1 \to 5$: $\big[\tfrac{u^4}{4}\big]_1^5 = \frac{625 - 1}{4} = 156$. C3. $u = 1 + \sqrt x$, $du = \tfrac{1}{2\sqrt x}dx$, limits $2 \to 3$: $2\big[\ln u\big]_2^3 = 2\ln\tfrac{3}{2}$. C5. $u = x^2 + 1$, limits $1 \to 2$: $\frac{1}{2}\ln 2$.

Part D (selected).

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D1. $u = x$, $dv = e^{2x}dx$, $v = \tfrac12 e^{2x}$: $\frac{x}{2}e^{2x} - \frac{1}{4}e^{2x} + C = \frac{1}{4}(2x-1)e^{2x} + C$. D3. $u = \ln(2x)$, $dv = dx$, $du = \tfrac{1}{x}dx$: $x\ln(2x) - x + C$. D5. $u = \arcsin x$, $dv = dx$; leftover $\int \tfrac{x}{\sqrt{1-x^2}}dx = -\sqrt{1-x^2}$: $x\arcsin x + \sqrt{1 - x^2} + C$. D7. Tabular ($x^2 \to 2x \to 2 \to 0$; integrals of $e^{-x}$: $-e^{-x}, e^{-x}, -e^{-x}$; signs $+,-,+$): $-x^2 e^{-x} - 2x e^{-x} - 2 e^{-x} + C = -(x^2 + 2x + 2)e^{-x} + C$. D9. $\big[-(x+1)e^{-x}\big]_0^1 = -2e^{-1} + 1 = 1 - \tfrac{2}{e}$.

Part E.

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E1. $I = \int e^{2x}\cos x\,dx$. Two passes (keep $u = e^{2x}$ both times) give $I = \tfrac{1}{5}e^{2x}(2\cos x + \sin x) + C$. Check: differentiating returns $e^{2x}\cos x$. E2. $\int e^{-x}\sin(2x)\,dx = -\tfrac{1}{5}e^{-x}(\sin 2x + 2\cos 2x) + C$.

Part F (selected).

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F1. $w = \sqrt x$, $x = w^2$, $dx = 2w\,dw$: $2\int w\sin w\,dw = 2(-w\cos w + \sin w)$, so $-2\sqrt x\cos\sqrt x + 2\sin\sqrt x + C$. F3. Two parts give $E[X^2] = \tfrac{2}{\lambda^2}$. Then $\operatorname{Var}(X) = \tfrac{2}{\lambda^2} - \tfrac{1}{\lambda^2} = \tfrac{1}{\lambda^2}$. F4. $W = \big[-(x+1)e^{-x}\big]_0^2 = 1 - 3e^{-2} \approx 0.594$ J. It is below $2e^{-1}\approx 0.736$ because $F$ rises then falls on $[0,2]$, peaking near $x=1$ but spending the right half decaying toward zero, so the midpoint value overstates the average force. F6. (a) $u$-sub, numerator is $\tfrac13$ the derivative of the denominator. (b) parts, log × algebraic. (c) $u$-sub, composite $\cos(x^2)$ with inner derivative $2x$ present up to a constant. (d) parts + rotating trick. (e) Chapter 16 trig substitution $x = 3\sin\theta$ — neither of this chapter's methods. (f) $u$-sub, numerator $2x+3$ is the derivative of the denominator.

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