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# 7.5E: Ejercicios para la Sección 7.5

• Edwin “Jed” Herman & Gilbert Strang
• OpenStax

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Utilice una tabla de integrales para evaluar las siguientes integrales.

1)$$\displaystyle ∫_0^4\frac{x}{\sqrt{1+2x}}\,dx$$

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

Contestar
$$\displaystyle ∫\frac{x+3}{x^2+2x+2}\,dx = \tfrac{1}{2}\ln |x^2+2x+2|+2\arctan(x+1)+C$$

3)$$\displaystyle ∫x^3\sqrt{1+2x^2}\,dx$$

4)$$\displaystyle ∫\frac{1}{\sqrt{x^2+6x}}\,dx$$

Contestar
$$\displaystyle ∫\frac{1}{\sqrt{x^2+6x}}\,dx = \cosh^{−1}\left(\frac{x+3}{3}\right)+C$$

5)$$\displaystyle ∫\frac{x}{x+1}\,dx$$

6)$$\displaystyle ∫x⋅2^{x^2}\,dx$$

Contestar
$$\displaystyle ∫x⋅2^{x^2}\,dx = \frac{2^{x^2−1}}{\ln 2}+C$$

7)$$\displaystyle ∫\frac{1}{4x^2+25}\,dx$$

8)$$\displaystyle ∫\frac{dy}{\sqrt{4−y^2}}$$

Contestar
$$\displaystyle ∫\frac{dy}{\sqrt{4−y^2}} = \arcsin\left(\frac{y}{2}\right)+C$$

9)$$\displaystyle ∫\sin^3(2x)\cos(2x)\,dx$$

10)$$\displaystyle ∫\csc(2w)\cot(2w)\,dw$$

Contestar
$$\displaystyle ∫\csc(2w)\cot(2w)\,dw = −\tfrac{1}{2}\csc(2w)+C$$

11)$$\displaystyle ∫2^y\,dy$$

12)$$\displaystyle ∫^1_0\frac{3x}{\sqrt{x^2+8}}\,dx$$

Contestar
$$\displaystyle ∫^1_0\frac{3x}{\sqrt{x^2+8}}\,dx = 9−6\sqrt{2}$$

13)$$\displaystyle ∫^{1/4}_{−1/4}\sec^2(πx)\tan(πx)\,dx$$

14)$$\displaystyle ∫^{π/2}_0\tan^2\left(\frac{x}{2}\right)\,dx$$

Contestar
$$\displaystyle ∫^{π/2}_0\tan^2\left(\frac{x}{2}\right)\,dx = 2−\frac{π}{2}$$

15)$$\displaystyle ∫\cos^3x\,dx$$

16)$$\displaystyle ∫\tan^5(3x)\,dx$$

Contestar
$$\displaystyle ∫\tan^5(3x)\,dx = \tfrac{1}{12}\tan^4(3x)−\tfrac{1}{6}\tan^2(3x)+\tfrac{1}{3}\ln|\sec 3x|+C$$

17)$$\displaystyle ∫\sin^2y\cos^3y\,dy$$

Utilice un CAS para evaluar las siguientes integrales. También se pueden usar tablas para verificar las respuestas.

18) [T]$$\displaystyle ∫\frac{dw}{1+\sec\left(\frac{w}{2}\right)}$$

Contestar
$$\displaystyle ∫\frac{dw}{1+\sec\left(\frac{w}{2}\right)} = 2\cot\left(\tfrac{w}{2}\right)−2\csc\left(\tfrac{w}{2}\right)+w+C$$

19) [T]$$\displaystyle ∫\frac{dw}{1−\cos(7w)}$$

20) [T]$$\displaystyle ∫^t_0\frac{dt}{4\cos t+3\sin t}$$

Contestar
$$\displaystyle ∫^t_0\frac{dt}{4\cos t+3\sin t} = \tfrac{1}{5}\ln\Big|\frac{2(5+4\sin t−3\cos t)}{4\cos t+3\sin t}\Big|$$

21) [T]$$\displaystyle ∫\frac{\sqrt{x^2−9}}{3x}\,dx$$

22) [T]$$\displaystyle ∫\frac{dx}{x^{1/2}+x^{1/3}}$$

Contestar
$$\displaystyle ∫\frac{dx}{x^{1/2}+x^{1/3}} = 6x^{1/6}−3x^{1/3}+2\sqrt{x}−6\ln[1+x^{1/6}]+C$$

23) [T]$$\displaystyle ∫\frac{dx}{x\sqrt{x−1}}$$

24) [T]$$\displaystyle ∫x^3\sin x\,dx$$

Contestar
$$\displaystyle ∫x^3\sin x\,dx = −x^3\cos x+3x^2\sin x+6x\cos x−6\sin x+C$$

25) [T]$$\displaystyle ∫x\sqrt{x^4−9}\,dx$$

26) [T]$$\displaystyle ∫\frac{x}{1+e^{−x^2}}\,dx$$

Contestar
$$\displaystyle ∫\frac{x}{1+e^{−x^2}}\,dx = \tfrac{1}{2}\left(x^2+\ln|1+e^{−x^2}|\right)+C$$

27) [T]$$\displaystyle ∫\frac{\sqrt{3−5x}}{2x}\,dx$$

28) [T]$$\displaystyle ∫\frac{dx}{x\sqrt{x−1}}$$

Contestar
$$\displaystyle ∫\frac{dx}{x\sqrt{x−1}} = 2\arctan\big(\sqrt{x−1}\big)+C$$

29) [T]$$\displaystyle ∫e^x\cos^{−1}(e^x)\,dx$$

Utilice una calculadora o CAS para evaluar las siguientes integrales.

30) [T]$$\displaystyle ∫^{π/4}_0\cos 2x \, dx$$

Contestar
$$\displaystyle ∫^{π/4}_0\cos 2x \, dx = 0.5=\frac{1}{2}$$

31) [T]$$\displaystyle ∫^1_0x⋅e^{−x^2}\,dx$$

32) [T]$$\displaystyle ∫^8_0\frac{2x}{\sqrt{x^2+36}}\,dx$$

Contestar
$$\displaystyle ∫^8_0\frac{2x}{\sqrt{x^2+36}}\,dx = 8.0$$

33) [T]$$\displaystyle ∫^{2/\sqrt{3}}_0\frac{1}{4+9x^2}\,dx$$

34) [T]$$\displaystyle ∫\frac{dx}{x^2+4x+13}$$

Contestar
$$\displaystyle ∫\frac{dx}{x^2+4x+13} = \tfrac{1}{3}\arctan\left(\tfrac{1}{3}(x+2)\right)+C$$

35) [T]$$\displaystyle ∫\frac{dx}{1+\sin x}$$

Usa tablas para evaluar las integrales. Es posible que deba completar el cuadrado o cambiar las variables para poner la integral en una forma dada en la tabla.

36)$$\displaystyle ∫\frac{dx}{x^2+2x+10}$$

Contestar
$$\displaystyle ∫\frac{dx}{x^2+2x+10} = \tfrac{1}{3}\arctan\left(\frac{x+1}{3}\right)+C$$

37)$$\displaystyle ∫\frac{dx}{\sqrt{x^2−6x}}$$

38)$$\displaystyle ∫\frac{e^x}{\sqrt{e^{2x}−4}}\,dx$$

Contestar
$$\displaystyle ∫\frac{e^x}{\sqrt{e^{2x}−4}}\,dx = \ln\left(e^x+\sqrt{4+e^{2x}}\right)+C$$

39)$$\displaystyle ∫\frac{\cos x}{\sin^2x+2\sin x}\,dx$$

40)$$\displaystyle ∫\frac{\arctan(x^3)}{x^4}\,dx$$

Contestar
$$\displaystyle ∫\frac{\arctan(x^3)}{x^4}\,dx = \ln x−\tfrac{1}{6}\ln(x^6+1)−\frac{\arctan(x^3)}{3x^3}+C$$

41)$$\displaystyle ∫\frac{\ln|x|\arcsin\left(\ln|x|\right)}{x}\,dx$$

Usar tablas para realizar la integración.

42)$$\displaystyle ∫\frac{dx}{\sqrt{x^2+16}}$$

Contestar
$$\displaystyle ∫\frac{dx}{\sqrt{x^2+16}} = \ln |x|+\sqrt{16+x^2}∣+C$$

43)$$\displaystyle ∫\frac{3x}{2x+7}\,dx$$

44)$$\displaystyle ∫\frac{dx}{1−\cos 4x}$$

Contestar
$$\displaystyle ∫\frac{dx}{1−\cos 4x} = −\frac{1}{4}\cot 2x+C$$

45)$$\displaystyle ∫\frac{dx}{\sqrt{4x+1}}$$

46) Encuentra el área delimitada por$$y(4+25x^2)=5,\;x=0,\;y=0,$$ y$$x=4.$$ Usa una tabla de integrales o un CAS.

Contestar
$$\frac{1}{2}\arctan 10$$unidades²

47) La región delimitada entre la curva$$y=\dfrac{1}{\sqrt{1+\cos x}}, \; 0.3≤x≤1.1,$$ y el$$x$$ eje -gira alrededor del$$x$$ eje para generar un sólido. Utilice una tabla de integrales para encontrar el volumen del sólido generado. (Redondear la respuesta a dos decimales.)

48) Utilice la sustitución y una tabla de integrales para encontrar el área de la superficie generada al girar la curva$$y=e^x,\; 0≤x≤3,$$ alrededor del$$x$$ eje -eje. (Redondear la respuesta a dos decimales.)

Contestar
$$1276.14$$unidades²

49) [T] Utilice una tabla integral y una calculadora para encontrar el área de la superficie generada al girar la curva$$y=\dfrac{x^2}{2},\; 0≤x≤1,$$ alrededor del$$x$$ eje -eje. (Redondear la respuesta a dos decimales.)

50) [T] Utilice un CAS o tablas para encontrar el área de la superficie generada al girar la curva$$y=\cos x,\; 0≤x≤\frac{π}{2},$$ alrededor del$$x$$ eje. (Redondear la respuesta a dos decimales.)

Contestar
$$7.21$$unidades²

51) Encuentra la longitud de la curva$$y=\dfrac{x^2}{4}$$ sobre$$[0,8]$$.

52) Encuentra la longitud de la curva$$y=e^x$$ sobre$$[0,\,\ln(2)].$$

Contestar
$$\left(\sqrt{5}−\sqrt{2}+\ln\Big|\frac{2+2\sqrt{2}}{1+\sqrt{5}}\Big|\right)$$unidades

53) Encontrar el área de la superficie formada girando la gráfica de$$y=2\sqrt{x}$$ sobre el intervalo$$[0,9]$$ alrededor del$$x$$ eje -eje.

54) Encuentra el valor promedio de la función$$f(x)=\dfrac{1}{x^2+1}$$ a lo largo del intervalo$$[−3,3].$$

Contestar
$$\frac{1}{3}\arctan(3)≈0.416$$

55) Aproximar la longitud del arco de la curva$$y=\tan πx$$ a lo largo del intervalo$$\left[0,\frac{1}{4}\right]$$. (Redondear la respuesta a tres decimales.)