In Problems $$ 27-60 $$, find each indefinite integral. $$ \begin{array}{ll} ext { (33) 27. } \int \sin (3 x) d x & ext { 28. } \int x \sin x^{2} d x \ ext { 29. } \int \sin x \cos ^{2} x d x & ext { 30. } \int an ^{2} x \sec ^{2} x d x \ ext { 31. } \int \frac{e^{1 / x}}{x^{2}} d x & ext { 32. } \int \frac{e^{\frac{3}{x}}}{\sqrt[3]{x^{2}}} d x \ ext { (435) 33. } \int \frac{x d x}{x^{2}-1} & ext { 34. } \int \frac{5 x d x}{1-x^{2}} \ ext { 35. } \int \frac{e^{x}}{\sqrt{1+e^{x}}} d x & ext { 36. } \int \frac{d x}{x(\ln x)^{7}}\end{array} $$

Question

In Problems $$ 27-60 $$, find each indefinite integral. $$ \begin{array}{ll}	ext { (33) 27. } \int \sin (3 x) d x & 	ext { 28. } \int x \sin x^{2} d x \ 	ext { 29. } \int \sin x \cos ^{2} x d x & 	ext { 30. } \int 	an ^{2} x \sec ^{2} x d x \ 	ext { 31. } \int \frac{e^{1 / x}}{x^{2}} d x & 	ext { 32. } \int \frac{e^{\frac{3}{x}}}{\sqrt[3]{x^{2}}} d x \ 	ext { (435) 33. } \int \frac{x d x}{x^{2}-1} & 	ext { 34. } \int \frac{5 x d x}{1-x^{2}} \ 	ext { 35. } \int \frac{e^{x}}{\sqrt{1+e^{x}}} d x & 	ext { 36. } \int \frac{d x}{x(\ln x)^{7}}\end{array} $$

UpStudy AI · Accepted Answer

Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \tan^{2}\left(x\right)\sec^{2}\left(x\right) dx$$
- step1: Multiply the terms:
  $$\int \left(\tan\left(x\right)\sec\left(x\right)\right)^{2} dx$$
- step2: Use the substitution $$dx=\frac{1}{\sec^{2}\left(x\right)} dt$$ to transform the integral$$:$$
  $$\int \left(\tan\left(x\right)\sec\left(x\right)\right)^{2}\times \frac{1}{\sec^{2}\left(x\right)} dt$$
- step3: Simplify:
  $$\int \tan^{2}\left(x\right) dt$$
- step4: Use the substitution $$t=\tan\left(x\right)$$ to transform the integral$$:$$
  $$\int t^{2} dt$$
- step5: Evaluate the integral:
  $$\frac{t^{2+1}}{2+1}$$
- step6: Add the numbers:
  $$\frac{t^{3}}{2+1}$$
- step7: Add the numbers:
  $$\frac{t^{3}}{3}$$
- step8: Substitute back:
  $$\frac{\tan^{3}\left(x\right)}{3}$$
- step9: Simplify the expression:
  $$\frac{1}{3}\tan^{3}\left(x\right)$$
- step10: Add the constant of integral C:
  $$\frac{1}{3}\tan^{3}\left(x\right) + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{dx}{x(\ln x)^{7}} $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{1}{x\left(\ln{\left(x\right)}\right)^{7}} dx$$
- step1: Simplify the expression:
  $$\int \frac{1}{\left(\ln{\left(x\right)}\right)^{7}x} dx$$
- step2: Use the substitution $$dx=x dt$$ to transform the integral$$:$$
  $$\int \frac{1}{\left(\ln{\left(x\right)}\right)^{7}x}\times x dt$$
- step3: Simplify:
  $$\int \frac{1}{\left(\ln{\left(x\right)}\right)^{7}} dt$$
- step4: Use the substitution $$t=\ln{\left(x\right)}$$ to transform the integral$$:$$
  $$\int \frac{1}{t^{7}} dt$$
- step5: Evaluate the integral:
  $$\frac{t^{-7+1}}{-7+1}$$
- step6: Add the numbers:
  $$\frac{t^{-6}}{-7+1}$$
- step7: Add the numbers:
  $$\frac{t^{-6}}{-6}$$
- step8: Rewrite the fraction:
  $$-\frac{t^{-6}}{6}$$
- step9: Rewrite the expression:
  $$-\frac{\frac{1}{t^{6}}}{6}$$
- step10: Simplify:
  $$-\frac{1}{6t^{6}}$$
- step11: Substitute back:
  $$-\frac{1}{6\left(\ln{\left(x\right)}\right)^{6}}$$
- step12: Add the constant of integral C:
  $$-\frac{1}{6\left(\ln{\left(x\right)}\right)^{6}} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{x}{x^{2}-1} dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{x}{x^{2}-1} dx$$
- step1: Use the substitution $$dx=\frac{1}{2x} dt$$ to transform the integral$$:$$
  $$\int \frac{x}{x^{2}-1}\times \frac{1}{2x} dt$$
- step2: Simplify:
  $$\int \frac{1}{2x^{2}-2} dt$$
- step3: Use the substitution $$t=x^{2}$$ to transform the integral$$:$$
  $$\int \frac{1}{2t-2} dt$$
- step4: Rewrite the expression:
  $$\int \frac{1}{2}\times \frac{1}{t-1} dt$$
- step5: Use properties of integrals:
  $$\frac{1}{2}\times \int \frac{1}{t-1} dt$$
- step6: Evaluate the integral:
  $$\frac{1}{2}\ln{\left(\left|t-1\right|\right)}$$
- step7: Substitute back:
  $$\frac{1}{2}\ln{\left(\left|x^{2}-1\right|\right)}$$
- step8: Add the constant of integral C:
  $$\frac{1}{2}\ln{\left(\left|x^{2}-1\right|\right)} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{e^{1/x}}{x^{2}} dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{e^{\frac{1}{x}}}{x^{2}} dx$$
- step1: Use the substitution $$dx=-x^{2} dt$$ to transform the integral$$:$$
  $$\int \frac{e^{\frac{1}{x}}}{x^{2}}\times \left(-x^{2}\right) dt$$
- step2: Simplify:
  $$\int -e^{\frac{1}{x}} dt$$
- step3: Use the substitution $$t=\frac{1}{x}$$ to transform the integral$$:$$
  $$\int -e^{t} dt$$
- step4: Use properties of integrals:
  $$-\int e^{t} dt$$
- step5: Evaluate the integral:
  $$-e^{t}$$
- step6: Substitute back:
  $$-e^{\frac{1}{x}}$$
- step7: Add the constant of integral C:
  $$-e^{\frac{1}{x}} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{5x}{1-x^{2}} dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{5x}{1-x^{2}} dx$$
- step1: Rewrite the expression:
  $$\int 5\times \frac{x}{1-x^{2}} dx$$
- step2: Use properties of integrals:
  $$5\times \int \frac{x}{1-x^{2}} dx$$
- step3: Use the substitution $$dx=-\frac{1}{2x} dt$$ to transform the integral$$:$$
  $$5\times \int \frac{x}{1-x^{2}}\times \left(-\frac{1}{2x}\right) dt$$
- step4: Simplify:
  $$5\times \int -\frac{1}{2+2\left(-x^{2}\right)} dt$$
- step5: Use the substitution $$t=-x^{2}$$ to transform the integral$$:$$
  $$5\times \int \frac{-1}{2+2t} dt$$
- step6: Simplify the expression:
  $$5\times \int -\frac{1}{2+2t} dt$$
- step7: Rewrite the expression:
  $$5\times \int -\frac{1}{2}\times \frac{1}{1+t} dt$$
- step8: Use properties of integrals:
  $$5\left(-\frac{1}{2}\right)\times \int \frac{1}{1+t} dt$$
- step9: Multiply the numbers:
  $$-\frac{5}{2}\times \int \frac{1}{1+t} dt$$
- step10: Evaluate the integral:
  $$-\frac{5}{2}\ln{\left(\left|t+1\right|\right)}$$
- step11: Substitute back:
  $$-\frac{5}{2}\ln{\left(\left|-x^{2}+1\right|\right)}$$
- step12: Calculate the absolute value:
  $$-\frac{5}{2}\ln{\left(\left|x^{2}-1\right|\right)}$$
- step13: Add the constant of integral C:
  $$-\frac{5}{2}\ln{\left(\left|x^{2}-1\right|\right)} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \sin(3x) dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \sin\left(3x\right) dx$$
- step1: Evaluate the integral:
  $$-\frac{1}{3}\cos\left(3x\right)$$
- step2: Add the constant of integral C:
  $$-\frac{1}{3}\cos\left(3x\right) + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{e^{x}}{\sqrt{1+e^{x}}} dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{e^{x}}{\sqrt{1+e^{x}}} dx$$
- step1: Rewrite the expression:
  $$\int \frac{e^{x}}{\left(1+e^{x}\right)^{\frac{1}{2}}} dx$$
- step2: Use the substitution $$dx=e^{-x} dt$$ to transform the integral$$:$$
  $$\int \frac{e^{x}}{\left(1+e^{x}\right)^{\frac{1}{2}}}\times e^{-x} dt$$
- step3: Simplify:
  $$\int \frac{1}{\left(1+e^{x}\right)^{\frac{1}{2}}} dt$$
- step4: Use the substitution $$t=e^{x}$$ to transform the integral$$:$$
  $$\int \frac{1}{\left(1+t\right)^{\frac{1}{2}}} dt$$
- step5: Use the substitution $$dt=1 dv$$ to transform the integral$$:$$
  $$\int \frac{1}{\left(1+t\right)^{\frac{1}{2}}}\times 1 dv$$
- step6: Simplify:
  $$\int \frac{1}{\left(1+t\right)^{\frac{1}{2}}} dv$$
- step7: Use the substitution $$v=1+t$$ to transform the integral$$:$$
  $$\int \frac{1}{v^{\frac{1}{2}}} dv$$
- step8: Evaluate the integral:
  $$\frac{v^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}$$
- step9: Add the numbers:
  $$\frac{v^{\frac{1}{2}}}{-\frac{1}{2}+1}$$
- step10: Add the numbers:
  $$\frac{v^{\frac{1}{2}}}{\frac{1}{2}}$$
- step11: Substitute back:
  $$\frac{\left(1+t\right)^{\frac{1}{2}}}{\frac{1}{2}}$$
- step12: Substitute back:
  $$\frac{\left(1+e^{x}\right)^{\frac{1}{2}}}{\frac{1}{2}}$$
- step13: Simplify:
  $$\frac{\sqrt{1+e^{x}}}{\frac{1}{2}}$$
- step14: Add the constant of integral C:
  $$\frac{\sqrt{1+e^{x}}}{\frac{1}{2}} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int x \sin(x^{2}) dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int x\sin\left(x^{2}\right) dx$$
- step1: Use the substitution $$dx=\frac{1}{2x} dt$$ to transform the integral$$:$$
  $$\int x\sin\left(x^{2}\right)\times \frac{1}{2x} dt$$
- step2: Simplify:
  $$\int \frac{\sin\left(x^{2}\right)}{2} dt$$
- step3: Use the substitution $$t=x^{2}$$ to transform the integral$$:$$
  $$\int \frac{\sin\left(t\right)}{2} dt$$
- step4: Simplify the expression:
  $$\int \frac{1}{2}\sin\left(t\right) dt$$
- step5: Use properties of integrals:
  $$\frac{1}{2}\times \int \sin\left(t\right) dt$$
- step6: Evaluate the integral:
  $$\frac{1}{2}\left(-\cos\left(t\right)\right)$$
- step7: Rewrite the expression:
  $$-\frac{1}{2}\cos\left(t\right)$$
- step8: Substitute back:
  $$-\frac{1}{2}\cos\left(x^{2}\right)$$
- step9: Add the constant of integral C:
  $$-\frac{1}{2}\cos\left(x^{2}\right) + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \sin(x) \cos^{2}(x) dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \sin\left(x\right)\cos^{2}\left(x\right) dx$$
- step1: Evaluate the integral:
  $$-\frac{\cos^{2+1}\left(x\right)}{2+1}$$
- step2: Add the numbers:
  $$-\frac{\cos^{3}\left(x\right)}{2+1}$$
- step3: Add the numbers:
  $$-\frac{\cos^{3}\left(x\right)}{3}$$
- step4: Simplify the expression:
  $$-\frac{1}{3}\cos^{3}\left(x\right)$$
- step5: Add the constant of integral C:
  $$-\frac{1}{3}\cos^{3}\left(x\right) + C, C \in \mathbb{R}$$
Here are the solutions for the indefinite integrals from the given problems:

1. **Problem 27:** 
   $$
   \int \sin(3x) \, dx = -\frac{1}{3} \cos(3x) + C, \quad C \in \mathbb{R}
   $$

2. **Problem 28:** 
   $$
   \int x \sin(x^{2}) \, dx = -\frac{1}{2} \cos(x^{2}) + C, \quad C \in \mathbb{R}
   $$

3. **Problem 29:** 
   $$
   \int \sin(x) \cos^{2}(x) \, dx = -\frac{1}{3} \cos^{3}(x) + C, \quad C \in \mathbb{R}
   $$

4. **Problem 30:** 
   $$
   \int \tan^{2}(x) \sec^{2}(x) \, dx = \frac{1}{3} \tan^{3}(x) + C, \quad C \in \mathbb{R}
   $$

5. **Problem 31:** 
   $$
   \int \frac{e^{1/x}}{x^{2}} \, dx = -e^{\frac{1}{x}} + C, \quad C \in \mathbb{R}
   $$

6. **Problem 32:** 
   The calculation for this integral was unsupported, so we cannot provide a solution.

7. **Problem 33:** 
   $$
   \int \frac{x}{x^{2}-1} \, dx = \frac{1}{2} \ln{(|x^{2}-1|)} + C, \quad C \in \mathbb{R}
   $$

8. **Problem 34:** 
   $$
   \int \frac{5x}{1-x^{2}} \, dx = -\frac{5}{2} \ln{(|x^{2}-1|)} + C, \quad C \in \mathbb{R}
   $$

9. **Problem 35:** 
   $$
   \int \frac{e^{x}}{\sqrt{1+e^{x}}} \, dx = 2\sqrt{1+e^{x}} + C, \quad C \in \mathbb{R}
   $$

10. **Problem 36:** 
    $$
    \int \frac{dx}{x(\ln x)^{7}} = -\frac{1}{6(\ln{(x)})^{6}} + C, \quad C \in \mathbb{R}
    $$

If you need further assistance or explanations for any specific integral, feel free to ask!
Here are the solutions for the indefinite integrals: 1. **Problem 27:** $$ \int \sin(3x) \, dx = -\frac{1}{3} \cos(3x) + C $$ 2. **Problem 28:** $$ \int x \sin(x^{2}) \, dx = -\frac{1}{2} \cos(x^{2}) + C $$ 3. **Problem 29:** $$ \int \sin(x) \cos^{2}(x) \, dx = -\frac{1}{3} \cos^{3}(x) + C $$ 4. **Problem 30:** $$ \int \tan^{2}(x) \sec^{2}(x) \, dx = \frac{1}{3} \tan^{3}(x) + C $$ 5. **Problem 31:** $$ \int \frac{e^{1/x}}{x^{2}} \, dx = -e^{\frac{1}{x}} + C $$ 6. **Problem 32:** Cannot solve this integral. 7. **Problem 33:** $$ \int \frac{x}{x^{2}-1} \, dx = \frac{1}{2} \ln{(|x^{2}-1|)} + C $$ 8. **Problem 34:** $$ \int \frac{5x}{1-x^{2}} \, dx = -\frac{5}{2} \ln{(|x^{2}-1|)} + C $$ 9. **Problem 35:** $$ \int \frac{e^{x}}{\sqrt{1+e^{x}}} \, dx = 2\sqrt{1+e^{x}} + C $$ 10. **Problem 36:** $$ \int \frac{dx}{x(\ln x)^{7}} = -\frac{1}{6(\ln{(x)})^{6}} + C $$ If you need more help with any specific integral, let me know!

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