39. $$ \int \frac{3 e^{x}}{\sqrt[4]{e^{x}-1}} d x $$ 41. $$ \int \frac{\cos x d x}{2 \sin x-1} $$ 64] 436. 43. $$ \int \sec (5 x) d x $$ 45. $$ \int \sqrt{ an x} \sec ^{2} x d x $$ 47. $$ \int \frac{\sin x}{\cos ^{2} x} d x $$ 49. $$ \int \sin x \cdot e^{\cos x} d x $$ 436. 51. $$ \int x \sqrt{x+3} d x $$ 53. $$ \int[\sin x+\cos (3 x)] d x $$ 4337 5 55. $$ \int \frac{d x}{x^{2}+25} $$ 57. $$ \int \frac{d x}{\sqrt{9-x^{2}}} $$ 59. $$ \int \sinh x \cosh x d x $$

Question

39. $$ \int \frac{3 e^{x}}{\sqrt[4]{e^{x}-1}} d x $$ 41. $$ \int \frac{\cos x d x}{2 \sin x-1} $$ 64] 436. 43. $$ \int \sec (5 x) d x $$ 45. $$ \int \sqrt{	an x} \sec ^{2} x d x $$ 47. $$ \int \frac{\sin x}{\cos ^{2} x} d x $$ 49. $$ \int \sin x \cdot e^{\cos x} d x $$ 436. 51. $$ \int x \sqrt{x+3} d x $$ 53. $$ \int[\sin x+\cos (3 x)] d x $$ 4337 5 55. $$ \int \frac{d x}{x^{2}+25} $$ 57. $$ \int \frac{d x}{\sqrt{9-x^{2}}} $$ 59. $$ \int \sinh x \cosh x d x $$

UpStudy AI · Accepted Answer

Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{\sin\left(x\right)}{\cos^{2}\left(x\right)} dx$$
- step1: Use the substitution $$dx=-\frac{1}{\sin\left(x\right)} dt$$ to transform the integral$$:$$
  $$\int \frac{\sin\left(x\right)}{\cos^{2}\left(x\right)}\times \left(-\frac{1}{\sin\left(x\right)}\right) dt$$
- step2: Simplify:
  $$\int -\frac{1}{\cos^{2}\left(x\right)} dt$$
- step3: Use the substitution $$t=\cos\left(x\right)$$ to transform the integral$$:$$
  $$\int \frac{-1}{t^{2}} dt$$
- step4: Rewrite the fraction:
  $$\int -\frac{1}{t^{2}} dt$$
- step5: Use properties of integrals:
  $$-\int \frac{1}{t^{2}} dt$$
- step6: Evaluate the integral:
  $$-\frac{t^{-2+1}}{-2+1}$$
- step7: Simplify:
  $$-\left(-\frac{1}{t}\right)$$
- step8: Calculate:
  $$\frac{1}{t}$$
- step9: Substitute back:
  $$\frac{1}{\cos\left(x\right)}$$
- step10: Rewrite the expression:
  $$\cos^{-1}\left(x\right)$$
- step11: Rewrite the expression:
  $$\sec\left(x\right)$$
- step12: Add the constant of integral C:
  $$\sec\left(x\right) + C, C \in \mathbb{R}$$
Calculate the integral $$ \int[\sin x+\cos (3 x)] d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \left(\sin\left(x\right)+\cos\left(3x\right)\right) dx$$
- step1: Use properties of integrals:
  $$\int \sin\left(x\right) dx+\int \cos\left(3x\right) dx$$
- step2: Evaluate the integral:
  $$-\cos\left(x\right)+\int \cos\left(3x\right) dx$$
- step3: Evaluate the integral:
  $$-\cos\left(x\right)+\frac{1}{3}\sin\left(3x\right)$$
- step4: Add the constant of integral C:
  $$-\cos\left(x\right)+\frac{1}{3}\sin\left(3x\right) + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \sin x \cdot e^{\cos x} d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \sin\left(x\right)\times e^{\cos\left(x\right)} dx$$
- step1: Use the substitution $$dx=-\frac{1}{\sin\left(x\right)} dt$$ to transform the integral$$:$$
  $$\int \sin\left(x\right)\times e^{\cos\left(x\right)}\left(-\frac{1}{\sin\left(x\right)}\right) dt$$
- step2: Simplify:
  $$\int -e^{\cos\left(x\right)} dt$$
- step3: Use the substitution $$t=\cos\left(x\right)$$ 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:
  $$-e^{\cos\left(x\right)}$$
- step7: Add the constant of integral C:
  $$-e^{\cos\left(x\right)} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{d x}{x^{2}+25} $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \frac{1}{x^{2}+25} dx$$
- step1: Evaluate the integral:
  $$\frac{1}{5}\arctan\left(\frac{x}{5}\right)$$
- step2: Add the constant of integral C:
  $$\frac{1}{5}\arctan\left(\frac{x}{5}\right) + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{\cos x d x}{2 \sin x-1} $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{\cos\left(x\right)}{2\sin\left(x\right)-1} dx$$
- step1: Use the substitution $$dx=\frac{1}{\cos\left(x\right)} dt$$ to transform the integral$$:$$
  $$\int \frac{\cos\left(x\right)}{2\sin\left(x\right)-1}\times \frac{1}{\cos\left(x\right)} dt$$
- step2: Simplify:
  $$\int \frac{1}{2\sin\left(x\right)-1} dt$$
- step3: Use the substitution $$t=\sin\left(x\right)$$ to transform the integral$$:$$
  $$\int \frac{1}{2t-1} dt$$
- step4: Evaluate the integral:
  $$\frac{1}{2}\ln{\left(\left|2t-1\right|\right)}$$
- step5: Substitute back:
  $$\frac{1}{2}\ln{\left(\left|2\sin\left(x\right)-1\right|\right)}$$
- step6: Add the constant of integral C:
  $$\frac{1}{2}\ln{\left(\left|2\sin\left(x\right)-1\right|\right)} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{d x}{\sqrt{9-x^{2}}} $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \frac{1}{\sqrt{9-x^{2}}} dx$$
- step1: Evaluate the integral:
  $$\arcsin\left(\frac{x}{3}\right)$$
- step2: Add the constant of integral C:
  $$\arcsin\left(\frac{x}{3}\right) + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \sec (5 x) d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \sec\left(5x\right) dx$$
- step1: Evaluate the integral:
  $$\frac{1}{5}\ln{\left(\left|\sec\left(5x\right)+\tan\left(5x\right)\right|\right)}$$
- step2: Add the constant of integral C:
  $$\frac{1}{5}\ln{\left(\left|\sec\left(5x\right)+\tan\left(5x\right)\right|\right)} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int x \sqrt{x+3} d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int x\sqrt{x+3} dx$$
- step1: Rewrite the expression:
  $$\int x\left(x+3\right)^{\frac{1}{2}} dx$$
- step2: Use the substitution $$dx=1 dt$$ to transform the integral$$:$$
  $$\int x\left(x+3\right)^{\frac{1}{2}}\times 1 dt$$
- step3: Simplify:
  $$\int \left(x+3\right)^{\frac{1}{2}}x dt$$
- step4: Use the substitution $$t=x+3$$ to transform the integral$$:$$
  $$\int t^{\frac{1}{2}}\left(t-3\right) dt$$
- step5: Multiply the terms:
  $$\int \left(t^{\frac{3}{2}}-3t^{\frac{1}{2}}\right) dt$$
- step6: Use properties of integrals:
  $$\int t^{\frac{3}{2}} dt+\int -3t^{\frac{1}{2}} dt$$
- step7: Evaluate the integral:
  $$\frac{2}{5}t^{\frac{5}{2}}+\int -3t^{\frac{1}{2}} dt$$
- step8: Evaluate the integral:
  $$\frac{2}{5}t^{\frac{5}{2}}-2t^{\frac{3}{2}}$$
- step9: Substitute back:
  $$\frac{2}{5}\left(x+3\right)^{\frac{5}{2}}-2\left(x+3\right)^{\frac{3}{2}}$$
- step10: Simplify:
  $$\frac{2}{5}\sqrt{\left(x+3\right)^{5}}-2\left(x+3\right)^{\frac{3}{2}}$$
- step11: Simplify:
  $$\frac{2}{5}\sqrt{\left(x+3\right)^{5}}-2\sqrt{\left(x+3\right)^{3}}$$
- step12: Expand the expression:
  $$\frac{2}{5}\left(x+3\right)^{2}\sqrt{x+3}+\left(-2x-6\right)\sqrt{x+3}$$
- step13: Expand the expression:
  $$\frac{2}{5}x^{2}\sqrt{x+3}+\frac{12}{5}x\sqrt{x+3}+\frac{18}{5}\sqrt{x+3}+\left(-2x-6\right)\sqrt{x+3}$$
- step14: Expand the expression:
  $$\frac{2}{5}x^{2}\sqrt{x+3}+\frac{12}{5}x\sqrt{x+3}+\frac{18}{5}\sqrt{x+3}-2x\sqrt{x+3}-6\sqrt{x+3}$$
- step15: Subtract the terms:
  $$\frac{2}{5}x^{2}\sqrt{x+3}+\frac{2}{5}\sqrt{x+3}\times x-\frac{12}{5}\sqrt{x+3}$$
- step16: Add the constant of integral C:
  $$\frac{2}{5}x^{2}\sqrt{x+3}+\frac{2}{5}\sqrt{x+3}\times x-\frac{12}{5}\sqrt{x+3} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{3 e^{x}}{\sqrt[4]{e^{x}-1}} d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{3e^{x}}{\sqrt[4]{e^{x}-1}} dx$$
- step1: Evaluate the power:
  $$\int \frac{3e^{x}}{\left(e^{x}-1\right)^{\frac{1}{4}}} dx$$
- step2: Rewrite the expression:
  $$\int 3\times \frac{e^{x}}{\left(e^{x}-1\right)^{\frac{1}{4}}} dx$$
- step3: Use properties of integrals:
  $$3\times \int \frac{e^{x}}{\left(e^{x}-1\right)^{\frac{1}{4}}} dx$$
- step4: Use the substitution $$dx=e^{-x} dt$$ to transform the integral$$:$$
  $$3\times \int \frac{e^{x}}{\left(e^{x}-1\right)^{\frac{1}{4}}}\times e^{-x} dt$$
- step5: Simplify:
  $$3\times \int \frac{1}{\left(e^{x}-1\right)^{\frac{1}{4}}} dt$$
- step6: Use the substitution $$t=e^{x}$$ to transform the integral$$:$$
  $$3\times \int \frac{1}{\left(t-1\right)^{\frac{1}{4}}} dt$$
- step7: Use the substitution $$dt=1 dv$$ to transform the integral$$:$$
  $$3\times \int \frac{1}{\left(t-1\right)^{\frac{1}{4}}}\times 1 dv$$
- step8: Simplify:
  $$3\times \int \frac{1}{\left(t-1\right)^{\frac{1}{4}}} dv$$
- step9: Use the substitution $$v=t-1$$ to transform the integral$$:$$
  $$3\times \int \frac{1}{v^{\frac{1}{4}}} dv$$
- step10: Evaluate the integral:
  $$3\times \frac{v^{-\frac{1}{4}+1}}{-\frac{1}{4}+1}$$
- step11: Simplify:
  $$3\times \frac{v^{\frac{3}{4}}}{\frac{3}{4}}$$
- step12: Multiply the terms:
  $$\frac{3v^{\frac{3}{4}}}{\frac{3}{4}}$$
- step13: Reduce the fraction:
  $$\frac{4v^{\frac{3}{4}}}{1}$$
- step14: Simplify:
  $$4v^{\frac{3}{4}}$$
- step15: Substitute back:
  $$4\left(t-1\right)^{\frac{3}{4}}$$
- step16: Substitute back:
  $$4\left(e^{x}-1\right)^{\frac{3}{4}}$$
- step17: Simplify:
  $$4\sqrt[4]{\left(e^{x}-1\right)^{3}}$$
- step18: Add the constant of integral C:
  $$4\sqrt[4]{\left(e^{x}-1\right)^{3}} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \sqrt{\tan x} \sec ^{2} x d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \sqrt{\tan\left(x\right)}\times \sec^{2}\left(x\right) dx$$
- step1: Rewrite the expression:
  $$\int \tan^{\frac{1}{2}}\left(x\right)\sec^{2}\left(x\right) dx$$
- step2: Use the substitution $$dx=\frac{1}{\sec^{2}\left(x\right)} dt$$ to transform the integral$$:$$
  $$\int \tan^{\frac{1}{2}}\left(x\right)\sec^{2}\left(x\right)\times \frac{1}{\sec^{2}\left(x\right)} dt$$
- step3: Simplify:
  $$\int \tan^{\frac{1}{2}}\left(x\right) dt$$
- step4: Use the substitution $$t=\tan\left(x\right)$$ to transform the integral$$:$$
  $$\int t^{\frac{1}{2}} dt$$
- step5: Evaluate the integral:
  $$\frac{t^{\frac{1}{2}+1}}{\frac{1}{2}+1}$$
- step6: Add the numbers:
  $$\frac{t^{\frac{3}{2}}}{\frac{1}{2}+1}$$
- step7: Add the numbers:
  $$\frac{t^{\frac{3}{2}}}{\frac{3}{2}}$$
- step8: Multiply by the reciprocal:
  $$t^{\frac{3}{2}}\times \frac{2}{3}$$
- step9: Reorder the terms:
  $$\frac{2}{3}t^{\frac{3}{2}}$$
- step10: Substitute back:
  $$\frac{2}{3}\tan^{\frac{3}{2}}\left(x\right)$$
- step11: Simplify:
  $$\frac{2}{3}\sqrt{\tan^{3}\left(x\right)}$$
- step12: Add the constant of integral C:
  $$\frac{2}{3}\sqrt{\tan^{3}\left(x\right)} + C, C \in \mathbb{R}$$
Here are the solutions to the integrals you provided:

1. **Integral**: $$ \int \frac{3 e^{x}}{\sqrt[4]{e^{x}-1}} d x $$
   - **Solution**: 
   $$
   4\sqrt[4]{(e^{x}-1)^{3}} + C, \quad C \in \mathbb{R}
   $$

2. **Integral**: $$ \int \frac{\cos x d x}{2 \sin x-1} $$
   - **Solution**: 
   $$
   \frac{1}{2}\ln{(|2\sin(x)-1|)} + C, \quad C \in \mathbb{R}
   $$

3. **Integral**: $$ \int \sec (5 x) d x $$
   - **Solution**: 
   $$
   \frac{1}{5}\ln{(|\sec(5x)+\tan(5x)|)} + C, \quad C \in \mathbb{R}
   $$

4. **Integral**: $$ \int \sqrt{\tan x} \sec ^{2} x d x $$
   - **Solution**: 
   $$
   \frac{2}{3}\sqrt{\tan^{3}(x)} + C, \quad C \in \mathbb{R}
   $$

5. **Integral**: $$ \int \frac{\sin x}{\cos ^{2} x} d x $$
   - **Solution**: 
   $$
   \sec(x) + C, \quad C \in \mathbb{R}
   $$

6. **Integral**: $$ \int \sin x \cdot e^{\cos x} d x $$
   - **Solution**: 
   $$
   -e^{\cos(x)} + C, \quad C \in \mathbb{R}
   $$

7. **Integral**: $$ \int x \sqrt{x+3} d x $$
   - **Solution**: 
   $$
   \frac{2}{5}x^{2}\sqrt{x+3} + \frac{2}{5}\sqrt{x+3}\times x - \frac{12}{5}\sqrt{x+3} + C, \quad C \in \mathbb{R}
   $$

8. **Integral**: $$ \int[\sin x+\cos (3 x)] d x $$
   - **Solution**: 
   $$
   -\cos(x) + \frac{1}{3}\sin(3x) + C, \quad C \in \mathbb{R}
   $$

9. **Integral**: $$ \int \frac{d x}{x^{2}+25} $$
   - **Solution**: 
   $$
   \frac{1}{5}\arctan\left(\frac{x}{5}\right) + C, \quad C \in \mathbb{R}
   $$

10. **Integral**: $$ \int \frac{d x}{\sqrt{9-x^{2}}} $$
    - **Solution**: 
    $$
    \arcsin\left(\frac{x}{3}\right) + C, \quad C \in \mathbb{R}
    $$

11. **Integral**: $$ \int \sinh x \cosh x d x $$
    - **Error**: The syntax of the input expression does not match the expected format.

If you need further assistance or corrections, please let me know!
Here are the solutions to the integrals: 1. $$ \int \frac{3 e^{x}}{\sqrt[4]{e^{x}-1}} d x = 4\sqrt[4]{(e^{x}-1)^{3}} + C $$ 2. $$ \int \frac{\cos x d x}{2 \sin x-1} = \frac{1}{2}\ln{(|2\sin(x)-1|)} + C $$ 3. $$ \int \sec (5 x) d x = \frac{1}{5}\ln{(|\sec(5x)+\tan(5x)|)} + C $$ 4. $$ \int \sqrt{\tan x} \sec ^{2} x d x = \frac{2}{3}\sqrt{\tan^{3}(x)} + C $$ 5. $$ \int \frac{\sin x}{\cos ^{2} x} d x = \sec(x) + C $$ 6. $$ \int \sin x \cdot e^{\cos x} d x = -e^{\cos(x)} + C $$ 7. $$ \int x \sqrt{x+3} d x = \frac{2}{5}x^{2}\sqrt{x+3} + \frac{2}{5}\sqrt{x+3}\times x - \frac{12}{5}\sqrt{x+3} + C $$ 8. $$ \int[\sin x+\cos (3 x)] d x = -\cos(x) + \frac{1}{3}\sin(3x) + C $$ 9. $$ \int \frac{d x}{x^{2}+25} = \frac{1}{5}\arctan\left(\frac{x}{5}\right) + C $$ 10. $$ \int \frac{d x}{\sqrt{9-x^{2}}} = \arcsin\left(\frac{x}{3}\right) + C $$ 11. $$ \int \sinh x \cosh x d x $$ has an error in the input expression. If you need further assistance or corrections, please let me know!

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