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# 5.5E & 5.6E U-Substitution Exercises

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## 5.5: Substitution

In the following exercises, find the antiderivative.

261) $$\displaystyle∫(x+1)^4\,dx$$

$$\displaystyle∫(x+1)^4\,dx \quad$$ $$=\quad \displaystyle\frac{1}{5}(x+1)^5+C$$

262) $$\displaystyle∫(x−1)^5\,dx$$

263) $$\displaystyle∫(2x−3)^{−7}\,dx$$

$$\displaystyle∫(2x−3)^{−7}\,dx\quad$$ $$=\quad\displaystyle−\frac{1}{12(3−2x)^6}+C$$

264) $$\displaystyle∫(3x−2)^{−11}\,dx$$

265) $$\displaystyle∫\frac{x}{\sqrt{x^2+1}}\,dx$$

$$\displaystyle∫\frac{x}{\sqrt{x^2+1}}\,dx\quad$$ $$=\quad \displaystyle\sqrt{x^2+1}+C$$

266) $$\displaystyle∫\frac{x}{\sqrt{1−x^2}}\,dx$$

267) $$\displaystyle∫(x−1)(x^2−2x)^3\,dx$$

$$\displaystyle∫(x−1)(x^2−2x)^3\,dx\quad$$ $$=\quad\displaystyle\frac{1}{8}(x^2−2x)^4+C$$

268) $$\displaystyle∫(x^2−2x)(x^3−3x^2)^2\,dx$$

269) $$\displaystyle\int\cos^3 θ\,dθ$$      (Hint: $$\cos^2 θ=1−\sin^2 θ$$)

$$\displaystyle\int\cos^3 θ\,dθ\quad$$ $$=\quad\displaystyle \sin θ−\frac{\sin^3 θ}{3}+C$$

270) $$\displaystyle\int\sin^3 θ\,dθ$$        (Hint: $$\sin^2 θ=1−\cos^2 θ$$)

271) $$\displaystyle\int x(1−x)^{99}\,dx$$

$$\displaystyle\int x(1−x)^{99}\,dx\quad$$ $$=\quad\displaystyle\frac{(1−x)^{101}}{101}−\frac{(1−x)^{100}}{100}+C$$

272) $$\displaystyle∫t(1−t^2)^{10}\,dt$$

273) $$\displaystyle∫(11x−7)^{−3}\,dx$$

$$\displaystyle−\frac{1}{22(7−11x^2)}+C$$

274) $$\displaystyle∫(7x−11)^4\,dx$$

275) $$\displaystyle\int\cos^3 θ\sin θ\,dθ$$

$$\displaystyle−\frac{\cos^4 θ}{4}+C$$

281) $$\displaystyle∫\frac{x^2}{(x^3−3)^2}\,dx$$

$$\displaystyle−\frac{1}{3(x^3−3)}+C$$

In the following exercises, evaluate the definite integral.

292) $$\displaystyle∫^1_0x\sqrt{1−x^2}\,dx$$

293) $$\displaystyle∫^1_0\frac{x}{\sqrt{1+x^2}}\,dx$$

$$\displaystyle u=1+x^2,du=2x\,dx,\frac{1}{2}∫^2_1u^{−1/2}du=\sqrt{2}−1$$

294) $$\displaystyle∫^2_0\frac{t}{\sqrt{5+t^2}}\,dt$$

295) $$\displaystyle∫^1_0\frac{t}{\sqrt{1+t^3}}\,dt$$

$$\displaystyle u=1+t^3,du=3t^2,\frac{1}{3}∫^2_1u^{−1/2}du=\frac{2}{3}(\sqrt{2}−1)$$

296) $$\displaystyle\int^{π/4}_0\sec^2 θ\tan θ\,dθ$$

297) $$\displaystyle\int^{π/4}_0\frac{\sin θ}{\cos^4 θ}\,dθ$$

$$\displaystyle u=\cos θ,\quad du=−\sin θ\,dθ,\quad∫^1_{1/\sqrt{2}}u^{−4}\,du=\frac{1}{3}(2\sqrt{2}−1)$$

## 5.6: Integrals Involving Exponential and Logarithmic Functions

In the following exercises, compute each indefinite integral.

320) $$\displaystyle ∫e^{2x}\,dx$$

321) $$\displaystyle ∫e^{−3x}\,dx$$

$$\displaystyle \frac{−1}{3}e^{−3x}+C$$

322) $$\displaystyle ∫2^x\,dx$$

323) $$\displaystyle ∫3^{−x}\,dx$$

$$\displaystyle −\frac{3^{−x}}{\ln 3}+C$$

324) $$\displaystyle ∫\frac{1}{2x}\,dx$$

325) $$\displaystyle ∫\frac{2}{x}\,dx$$

$$\displaystyle \ln(x^2)+C$$

326) $$\displaystyle ∫\frac{1}{x^2}\,dx$$

327) $$\displaystyle ∫\frac{1}{\sqrt{x}}\,dx$$

$$\displaystyle 2\sqrt{x}+C$$

In the following exercises, find each indefinite integral by using appropriate substitutions.

328) $$\displaystyle ∫\frac{\ln x}{x}\,dx$$

329) $$\displaystyle ∫\frac{\,dx}{x(\ln x)^2}$$

$$\displaystyle −\frac{1}{\ln x}+C$$

336) $$\displaystyle ∫xe^{−x^2}\,dx$$

337) $$\displaystyle ∫x^2e^{−x^3}\,dx$$

$$\displaystyle \frac{−e^{−x^3}}{3}+C$$

338) $$\displaystyle ∫e^{\sin x}\cos x\,dx$$

339) $$\displaystyle ∫e^{\tan x}\sec^2 x\,dx$$

$$\displaystyle e^{\tan x}+C$$

340) $$\displaystyle ∫\frac{e^{\ln x}}{x}\,dx$$

341) $$\displaystyle ∫\frac{e^{\ln(1−t)}}{1−t}\,dt$$

$$\displaystyle t+C$$

In the following exercises, evaluate the definite integral.

355) $$\displaystyle ∫^2_1\frac{1+2x+x^2}{3x+3x^2+x^3}\,dx$$

$$\displaystyle \frac{1}{3}\ln(\frac{26}{7})$$

356) $$\displaystyle ∫^{π/4}_0\tan x\,dx$$

357) $$\displaystyle ∫^{π/3}_0\frac{\sin x−\cos x}{\sin x+\cos x}\,dx$$

$$\displaystyle ln(\sqrt{3}−1)$$

358) $$\displaystyle ∫^{π/2}_{π/6}\csc x\,dx$$

359) $$\displaystyle ∫^{π/3}_{π/4}\cot x\,dx$$

$$\displaystyle \frac{1}{2}\ln\frac{3}{2}$$

In the following exercises, integrate using the indicated substitution.

360) $$\displaystyle ∫\frac{x}{x−100}\,dx;\quad u=x−100$$

361) $$\displaystyle ∫\frac{y−1}{y+1}dy;\quad u=y+1$$

$$\displaystyle y−2\ln|y+1|+C$$

362) $$\displaystyle ∫\frac{1−x^2}{3x−x^3}\,dx;\quad u=3x−x^3$$

363) $$\displaystyle ∫\frac{\sin x+\cos x}{\sin x−\cos x}\,dx;\quad u=\sin x−\cos x$$

$$\displaystyle \ln|\sin x−\cos x|+C$$

364) $$\displaystyle ∫e^{2x}\sqrt{1−e^{2x}}\,dx;\quad u=e^{2x}$$

365) $$\displaystyle ∫\ln(x)\frac{\sqrt{1−(\ln x)^2}}{x}\,dx;\quad u=\ln x$$

$$\displaystyle −\frac{1}{3}(1−(\ln x^2))^{3/2}+C$$

In the following exercises, $$\displaystyle f(x)≥0$$ for $$\displaystyle a≤x≤b$$. Find the area under the graph of $$\displaystyle f(x)$$ between the given values $$a$$ and $$b$$ by integrating.

372) $$\displaystyle f(x)=\frac{\log_{10}(x)}{x};\quad a=10,b=100$$

373) $$\displaystyle f(x)=\frac{\log_2(x)}{x};\quad a=32,b=64$$

$$\displaystyle \frac{11}{2}\ln 2$$

374) $$\displaystyle f(x)=2^{−x};\quad a=1,b=2$$

375) $$\displaystyle f(x)=2^{−x};\quad a=3,b=4$$

$$\displaystyle \frac{1}{\ln(65,536)}$$

376) Find the area under the graph of the function $$\displaystyle f(x)=xe^{−x^2}$$ between $$\displaystyle x=0$$ and $$\displaystyle x=5$$.

377) Compute the integral of $$\displaystyle f(x)=xe^{−x^2}$$ and find the smallest value of N such that the area under the graph $$\displaystyle f(x)=xe^{−x^2}$$ between $$\displaystyle x=N$$ and $$\displaystyle x=N+10$$ is, at most, 0.01.

$$\displaystyle ∫^{N+1}_Nxe^{−x^2}\,dx=\frac{1}{2}(e^{−N^2}−e^{−(N+1)^2}).$$ The quantity is less than 0.01 when $$\displaystyle N=2$$.

378) Find the limit, as N tends to infinity, of the area under the graph of $$\displaystyle f(x)=xe^{−x^2}$$ between $$\displaystyle x=0$$ and $$\displaystyle x=5$$.

379) Show that $$\displaystyle ∫^b_a\frac{\,dt}{t}=∫^{1/a}_{1/b}\frac{\,dt}{t}$$ when $$\displaystyle 0<a≤b$$.

$$\displaystyle ∫^b_a\frac{\,dx}{x}=\ln(b)−\ln(a)=\ln(\frac{1}{a})−\ln(\frac{1}{b})=∫^{1/a}_{1/b}\frac{\,dx}{x}$$

380) Suppose that $$\displaystyle f(x)>0$$ for all x and that f and g are differentiable. Use the identity $$\displaystyle f^g=e^{glnf}$$ and the chain rule to find the derivative of $$\displaystyle f^g$$.

381) Use the previous exercise to find the antiderivative of $$\displaystyle h(x)=x^x(1+\ln x)$$ and evaluate $$\displaystyle ∫^3_2x^x(1+\ln x)\,dx$$.

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382) Show that if $$\displaystyle c>0$$, then the integral of $$\displaystyle 1/x$$ from ac to bc $$\displaystyle (0<a<b)$$ is the same as the integral of $$\displaystyle 1/x$$ from a to b.

The following exercises are intended to derive the fundamental properties of the natural log starting from the definition $$\displaystyle \ln(x)=∫^x_1\frac{\,dt}{t}$$, using properties of the definite integral and making no further assumptions.

383) Use the identity $$\displaystyle \ln(x)=∫^x_1\frac{\,dt}{t}$$ to derive the identity $$\displaystyle \ln(\frac{1}{x})=−\ln x$$.

Solution: We may assume that $$\displaystyle x>1$$,so $$\displaystyle \frac{1}{x}<1.$$ Then, $$\displaystyle ∫^{1/x}_{1}\frac{\,dt}{t}$$. Now make the substitution $$\displaystyle u=\frac{1}{t}$$, so $$\displaystyle du=−\frac{\,dt}{t^2}$$ and $$\displaystyle \frac{du}{u}=−\frac{\,dt}{t}$$, and change endpoints: $$\displaystyle ∫^{1/x}_1\frac{\,dt}{t}=−∫^x_1\frac{du}{u}=−lnx.$$

384) Use a change of variable in the integral $$\displaystyle ∫^{xy}_1\frac{1}{t}\,dt$$ to show that $$\displaystyle lnxy=lnx+lny$$ for $$\displaystyle x,y>0$$.

385) Use the identity $$\displaystyle \ln x=∫^x_1\frac{\,dt}{x}$$ to show that $$\displaystyle \ln(x)$$ is an increasing function of $$x$$ on $$\displaystyle [0,∞)$$, and use the previous exercises to show that the range of $$\displaystyle \ln(x)$$ is $$\displaystyle (−∞,∞)$$. Without any further assumptions, conclude that $$\displaystyle \ln(x)$$ has an inverse function defined on $$\displaystyle (−∞,∞).$$

386) Pretend, for the moment, that we do not know that $$\displaystyle e^x$$ is the inverse function of $$\displaystyle \ln(x)$$, but keep in mind that $$\displaystyle \ln(x)$$ has an inverse function defined on $$\displaystyle (−∞,∞)$$. Call it $$E$$. Use the identity $$\displaystyle \ln xy=\ln x+\ln y$$ to deduce that $$\displaystyle E(a+b)=E(a)E(b)$$ for any real numbers $$a$$, $$b$$.

387) Pretend, for the moment, that we do not know that $$\displaystyle e^x$$ is the inverse function of $$\displaystyle \ln x$$, but keep in mind that $$\displaystyle \ln x$$ has an inverse function defined on $$\displaystyle (−∞,∞)$$. Call it $$E$$. Show that $$\displaystyle E'(t)=E(t).$$

Solution: $$\displaystyle x=E(\ln(x)).$$ Then, $$\displaystyle 1=\frac{E'(\ln x)}{x}$$ or $$\displaystyle x=E'(\ln x)$$. Since any number t can be written $$\displaystyle t=\ln x$$ for some $$x$$, and for such $$t$$ we have $$\displaystyle x=E(t)$$, it follows that for any $$\displaystyle t,\,E'(t)=E(t).$$

388) The sine integral, defined as $$\displaystyle S(x)=∫^x_0\frac{\sin t}{t}\,dt$$ is an important quantity in engineering. Although it does not have a simple closed formula, it is possible to estimate its behavior for large $$x$$. Show that for $$\displaystyle k≥1,|S(2πk)−S(2π(k+1))|≤\frac{1}{k(2k+1)π}.$$ (Hint: $$\displaystyle \sin(t+π)=−\sin t$$)

389) [T] The normal distribution in probability is given by $$\displaystyle p(x)=\frac{1}{σ\sqrt{2π}}e^{−(x−μ)^2/2σ^2}$$, where $$σ$$ is the standard deviation and $$μ$$ is the average. The standard normal distribution in probability, $$\displaystyle p_s$$, corresponds to $$\displaystyle μ=0$$ and $$\displaystyle σ=1$$. Compute the left endpoint estimates $$\displaystyle R_{10}$$ and $$\displaystyle R_{100}$$ of $$\displaystyle ∫^1_{−1}\frac{1}{\sqrt{2π}}e^{−x^{2/2}}\,dx.$$

Solution: $$\displaystyle R_{10}=0.6811,R_{100}=0.6827$$

390) [T] Compute the right endpoint estimates $$\displaystyle R_{50}$$ and $$\displaystyle R_{100}$$ of $$\displaystyle ∫^5_{−3}\frac{1}{2\sqrt{2π}}e^{−(x−1)^2/8}$$.