In Problems 69-78, find each definit 71. $$ \int_{0}^{2} \frac{e^{2 x}}{e^{2 x}+1} d x $$ 73. $$ \int_{2}^{3} \frac{d x}{x \ln x} $$ 75. $$ \int_{0}^{\pi} e^{x} \cos \left(e^{x}\right) d x $$ 77. $$ \int_{0}^{1} \frac{x d x}{1+x^{4}} $$

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Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int_{0}^{1} \frac{x}{1+x^{4}} dx$$
- step1: Evaluate the integral:
  $$\int \frac{x}{1+x^{4}} dx$$
- step2: Use the substitution $$dx=\frac{1}{2x} dt$$ to transform the integral$$:$$
  $$\int \frac{x}{1+x^{4}}\times \frac{1}{2x} dt$$
- step3: Simplify:
  $$\int \frac{1}{2+2\left(x^{2}\right)^{2}} dt$$
- step4: Use the substitution $$t=x^{2}$$ to transform the integral$$:$$
  $$\int \frac{1}{2+2t^{2}} dt$$
- step5: Rewrite the expression:
  $$\int \frac{1}{2}\times \frac{1}{1+t^{2}} dt$$
- step6: Use properties of integrals:
  $$\frac{1}{2}\times \int \frac{1}{1+t^{2}} dt$$
- step7: Evaluate the integral:
  $$\frac{1}{2}\arctan\left(t\right)$$
- step8: Substitute back:
  $$\frac{1}{2}\arctan\left(x^{2}\right)$$
- step9: Return the limits:
  $$\left(\frac{1}{2}\arctan\left(x^{2}\right)\right)\bigg |_{0}^{1}$$
- step10: Calculate the value:
  $$\frac{\pi }{8}$$
Calculate the integral $$ \int_{2}^{3} \frac{d x}{x \ln x} $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int_{2}^{3} \frac{1}{x\ln{\left(x\right)}} dx$$
- step1: Evaluate the integral:
  $$\int \frac{1}{x\ln{\left(x\right)}} dx$$
- step2: Use the substitution $$dx=x dt$$ to transform the integral$$:$$
  $$\int \frac{1}{x\ln{\left(x\right)}}\times x dt$$
- step3: Simplify:
  $$\int \frac{1}{\ln{\left(x\right)}} dt$$
- step4: Use the substitution $$t=\ln{\left(x\right)}$$ to transform the integral$$:$$
  $$\int \frac{1}{t} dt$$
- step5: Evaluate the integral:
  $$\ln{\left(\left|t\right|\right)}$$
- step6: Substitute back:
  $$\ln{\left(\left|\ln{\left(x\right)}\right|\right)}$$
- step7: Return the limits:
  $$\left(\ln{\left(\left|\ln{\left(x\right)}\right|\right)}\right)\bigg |_{2}^{3}$$
- step8: Calculate the value:
  $$\ln{\left(\log_{2}{\left(3\right)}\right)}$$
Calculate the integral $$ \int_{0}^{\pi} e^{x} \cos \left(e^{x}\right) d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int_{0}^{\pi } e^{x}\cos\left(e^{x}\right) dx$$
- step1: Evaluate the integral:
  $$\int e^{x}\cos\left(e^{x}\right) dx$$
- step2: Use the substitution $$dx=e^{-x} dt$$ to transform the integral$$:$$
  $$\int e^{x}\cos\left(e^{x}\right)\times e^{-x} dt$$
- step3: Use the substitution $$t=e^{x}$$ to transform the integral$$:$$
  $$\int t\cos\left(t\right)\times t^{-1} dt$$
- step4: Simplify the expression:
  $$\int \cos\left(t\right) dt$$
- step5: Evaluate the integral:
  $$\sin\left(t\right)$$
- step6: Substitute back:
  $$\sin\left(e^{x}\right)$$
- step7: Return the limits:
  $$\left(\sin\left(e^{x}\right)\right)\bigg |_{0}^{\pi }$$
- step8: Calculate the value:
  $$\sin\left(e^{\pi }\right)-\sin\left(1\right)$$
Calculate the integral $$ \int_{0}^{2} \frac{e^{2 x}}{e^{2 x}+1} d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int_{0}^{2} \frac{e^{2x}}{e^{2x}+1} dx$$
- step1: Evaluate the integral:
  $$\int \frac{e^{2x}}{e^{2x}+1} dx$$
- step2: Use the substitution $$dx=\frac{1}{2e^{2x}} dt$$ to transform the integral$$:$$
  $$\int \frac{e^{2x}}{e^{2x}+1}\times \frac{1}{2e^{2x}} dt$$
- step3: Simplify:
  $$\int \frac{1}{2e^{2x}+2} dt$$
- step4: Use the substitution $$t=e^{2x}$$ to transform the integral$$:$$
  $$\int \frac{1}{2t+2} dt$$
- step5: Rewrite the expression:
  $$\int \frac{1}{2}\times \frac{1}{t+1} dt$$
- step6: Use properties of integrals:
  $$\frac{1}{2}\times \int \frac{1}{t+1} dt$$
- step7: Evaluate the integral:
  $$\frac{1}{2}\ln{\left(\left|t+1\right|\right)}$$
- step8: Substitute back:
  $$\frac{1}{2}\ln{\left(\left|e^{2x}+1\right|\right)}$$
- step9: Calculate the absolute value:
  $$\frac{1}{2}\ln{\left(e^{2x}+1\right)}$$
- step10: Return the limits:
  $$\left(\frac{1}{2}\ln{\left(e^{2x}+1\right)}\right)\bigg |_{0}^{2}$$
- step11: Calculate the value:
  $$\frac{\ln{\left(\frac{e^{4}+1}{2}\right)}}{2}$$
Here are the results for each of the integrals:

71. For the integral $$ \int_{0}^{2} \frac{e^{2 x}}{e^{2 x}+1} d x $$:
   $$
   \int_{0}^{2} \frac{e^{2 x}}{e^{2 x}+1} d x = \frac{\ln{\left(\frac{e^{4}+1}{2}\right)}}{2}
   $$

73. For the integral $$ \int_{2}^{3} \frac{d x}{x \ln x} $$:
   $$
   \int_{2}^{3} \frac{d x}{x \ln x} = \ln{(\log_{2}{(3)})}
   $$

75. For the integral $$ \int_{0}^{\pi} e^{x} \cos \left(e^{x}\right) d x $$:
   $$
   \int_{0}^{\pi} e^{x} \cos \left(e^{x}\right) d x = \sin(e^{\pi}) - \sin(1)
   $$

77. For the integral $$ \int_{0}^{1} \frac{x d x}{1+x^{4}} $$:
   $$
   \int_{0}^{1} \frac{x d x}{1+x^{4}} = \frac{\pi}{8}
   $$

If you have any further questions or need additional assistance, feel free to ask!
Here are the results for each integral: 71. $$ \int_{0}^{2} \frac{e^{2 x}}{e^{2 x}+1} d x = \frac{\ln{\left(\frac{e^{4}+1}{2}\right)}}{2} $$ 73. $$ \int_{2}^{3} \frac{d x}{x \ln x} = \ln{(\log_{2}{(3)})} $$ 75. $$ \int_{0}^{\pi} e^{x} \cos \left(e^{x}\right) d x = \sin(e^{\pi}) - \sin(1) $$ 77. $$ \int_{0}^{1} \frac{x d x}{1+x^{4}} = \frac{\pi}{8} $$

In Problems 69-78, find each definit
71.
73.
75.
77.

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In Problems 69-78, find each definit 71. 73. 75. 77.