# 5.10: 5.5E and 5.6E U-Substitution Exercises

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## 5.5: Substitution

In the following exercises, find the antiderivative.

261) $$\displaystyle∫(x+1)^4dx$$

$$\displaystyle\frac{1}{5}(x+1)^5+C$$

262) $$\displaystyle∫(x−1)^5dx$$

263) $$\displaystyle∫(2x−3)^{−7}dx$$

$$\displaystyle−\frac{1}{12(3−2x)^6}+C$$

264) $$\displaystyle∫(3x−2)^{−11}dx$$

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

$$\displaystyle\sqrt{x^2+1}+C$$

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

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

$$\displaystyle\frac{1}{8}(x^2−2x)^4+C$$

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

269) $$\displaystyle∫cos^3θdθ(Hint:cos^2θ=1−sin^2θ)$$

$$\displaystylesinθ−\frac{sin^3θ}{3}+C$$

270) $$\displaystyle∫sin^3θdθ(Hint:sin^2θ=1−cos^2θ)$$

271) $$\displaystyle∫x(1−x)^{99}dx$$

Solution: $$\displaystyle\frac{(1−x)^{101}}{101}−\frac{(1−x)^{100}}{100}+C$$

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

273) $$\displaystyle∫(11x−7)^{−3}dx$$

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

274) $$\displaystyle∫(7x−11)^4dx$$

275) $$\displaystyle∫cos^3θsinθdθ$$

$$\displaystyle−\frac{cos^4θ}{4}+C$$
276) $$\displaystyle∫sin^3θdθ;u=cosθ (Hint:sin^2θ=1−cos^2θ)$$

(removed exercises 277- 280)

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

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

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) $$\displaystyle∫^{π/4}_0sec^2θtanθdθ$$

297) $$\displaystyle∫^{π/4}_0\frac{sinθ}{cos^4θ}dθ$$

$$\displaystyle u=cosθ,du=−sinθdθ,∫^1_{1/\sqrt{2}}u^{−4}du=\frac{1}{3}(2\sqrt{2}−1)$$

J5.5.1)

J5.5.2)

## 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) $$\displaystyle ∫2^xdx$$

323) $$\displaystyle ∫3^{−x}dx$$

$$\displaystyle −\frac{3^{−x}}{ln3}+C$$

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

325) $$\displaystyle ∫\frac{2}{x}dx$$

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

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) $$\displaystyle ∫\frac{lnx}{x}dx$$

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

$$\displaystyle −\frac{1}{lnx}+C$$

336) $$\displaystyle ∫xe^{−x^2}dx$$

337) $$\displaystyle ∫x^2e^{−x^3}dx$$

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

338) $$\displaystyle ∫e^{sinx}cosxdx$$

339) $$\displaystyle ∫e^{tanx}sec^2xdx$$

$$\displaystyle e^{tanx}+C$$

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

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) $$\displaystyle ∫^{π/4}_0tanxdx$$

357) $$\displaystyle ∫^{π/3}_0\frac{sinx−cosx}{sinx+cosx}dx$$

$$\displaystyle ln(\sqrt{3}−1)$$

358) $$\displaystyle ∫^{π/2}_{π/6}cscxdx$$

359) $$\displaystyle ∫^{π/3}_{π/4}cotxdx$$

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

In the following exercises, integrate using the indicated substitution.

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

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

$$\displaystyle y−2ln|y+1|+C$$

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

363) $$\displaystyle ∫\frac{sinx+cosx}{sinx−cosx}dx;u=sinx−cosx$$

$$\displaystyle ln|sinx−cosx|+C$$

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

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

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

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

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

$$\displaystyle \frac{11}{2}ln2$$

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

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

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

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 $$\displaystyle h(x)=x^x(1+lnx)$$ and evaluate $$\displaystyle ∫^3_2x^x(1+lnx)dx$$.

23

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 $$\displaystyle ln(\frac{1}{x})=−lnx$$.

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 $$\displaystyle lnx=∫^x_1\frac{dt}{x}$$ 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 $$\displaystyle lnxy=lnx+lny$$ 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 $$\displaystyle lnx$$, but keep in mind that $$\displaystyle lnx$$ has an inverse function defined on $$\displaystyle (−∞,∞)$$. Call it E. Show that $$\displaystyle E'(t)=E(t).$$

Solution: $$\displaystyle x=E(ln(x)).$$ Then, $$\displaystyle 1=\frac{E'(lnx)}{x}$$ or $$\displaystyle x=E'(lnx)$$. Since any number t can be written $$\displaystyle t=lnx$$ 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 $$\displaystyle S(x)=∫^x_0\frac{sint}{t}dt$$ 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: $$\displaystyle sin(t+π)=−sint$$)

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 $$\displaystyle ∫^5_{−3}\frac{1}{2\sqrt{2π}}e^{−(x−1)^2/8}$$.

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