In Problems 69-78, find each defi 71. $$ \int_{0}^{2} \frac{e^{2 x}}{e^{2 x}+1} d x $$ 73. $$ \int_{1}^{3} \frac{1}{x^{2}} \sqrt{1-\frac{1}{x}} d x $$ 75. $$ \int_{0}^{3} \frac{d x}{x \ln x} $$ 77. $$ \int_{0}^{\pi} \frac{x d x}{1+x^{4}} $$

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Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int_{0}^{\pi } \frac{x}{1+x^{4}} dx$$
- 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:
  $$\left(\frac{1}{2}\arctan\left(x^{2}\right)\right)\bigg |_{0}^{\pi }$$
- step10: Calculate the value:
  $$\frac{\arctan\left(\pi ^{2}\right)}{2}$$
Calculate the integral $$ \int_{1}^{3} \frac{1}{x^{2}} \sqrt{1-\frac{1}{x}} d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int_{1}^{3} \frac{1}{x^{2}}\times \sqrt{1-\frac{1}{x}} dx$$
- step1: Multiply the terms:
  $$\int_{1}^{3} \frac{\sqrt{1-\frac{1}{x}}}{x^{2}} dx$$
- step2: Evaluate the integral:
  $$\int \frac{\sqrt{1-\frac{1}{x}}}{x^{2}} dx$$
- step3: Rewrite the expression:
  $$\int \frac{\left(1-\frac{1}{x}\right)^{\frac{1}{2}}}{x^{2}} dx$$
- step4: Use the substitution $$dx=x^{2} dt$$ to transform the integral$$:$$
  $$\int \frac{\left(1-\frac{1}{x}\right)^{\frac{1}{2}}}{x^{2}}\times x^{2} dt$$
- step5: Simplify:
  $$\int \left(1-\frac{1}{x}\right)^{\frac{1}{2}} dt$$
- step6: Use the substitution $$t=-\frac{1}{x}$$ to transform the integral$$:$$
  $$\int \left(1+t\right)^{\frac{1}{2}} dt$$
- step7: Use the substitution $$dt=1 dv$$ to transform the integral$$:$$
  $$\int \left(1+t\right)^{\frac{1}{2}}\times 1 dv$$
- step8: Simplify:
  $$\int \left(1+t\right)^{\frac{1}{2}} dv$$
- step9: Use the substitution $$v=1+t$$ to transform the integral$$:$$
  $$\int v^{\frac{1}{2}} dv$$
- step10: Evaluate the integral:
  $$\frac{v^{\frac{1}{2}+1}}{\frac{1}{2}+1}$$
- step11: Add the numbers:
  $$\frac{v^{\frac{3}{2}}}{\frac{1}{2}+1}$$
- step12: Add the numbers:
  $$\frac{v^{\frac{3}{2}}}{\frac{3}{2}}$$
- step13: Substitute back:
  $$\frac{\left(1+t\right)^{\frac{3}{2}}}{\frac{3}{2}}$$
- step14: Substitute back:
  $$\frac{\left(1-\frac{1}{x}\right)^{\frac{3}{2}}}{\frac{3}{2}}$$
- step15: Return the limits:
  $$\left(\frac{\left(1-\frac{1}{x}\right)^{\frac{3}{2}}}{\frac{3}{2}}\right)\bigg |_{1}^{3}$$
- step16: Calculate the value:
  $$\frac{4\sqrt{6}}{27}$$
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:
  $$\frac{\ln{\left(\frac{e^{4}+1}{2}\right)}}{2}$$
Calculate the integral $$ \int_{0}^{3} \frac{d x}{x \ln x} $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int_{0}^{3} \frac{1}{x\ln{\left(x\right)}} dx$$
- step1: Transform the expression:
  $$\int_{0}^{\frac{1}{2}} \frac{1}{x\ln{\left(x\right)}} dx+\int_{\frac{1}{2}}^{1} \frac{1}{x\ln{\left(x\right)}} dx+\int_{1}^{3} \frac{1}{x\ln{\left(x\right)}} dx$$
- step2: Rewrite the improper integral:
  $$\lim _{a\rightarrow 0^{+}}\left(\int_{a}^{\frac{1}{2}} \frac{1}{x\ln{\left(x\right)}} dx\right)+\lim _{a\rightarrow 1^{-}}\left(\int_{\frac{1}{2}}^{a} \frac{1}{x\ln{\left(x\right)}} dx\right)+\lim _{a\rightarrow 1^{+}}\left(\int_{a}^{3} \frac{1}{x\ln{\left(x\right)}} dx\right)$$
- step3: Evaluate the integral:
  $$\lim _{a\rightarrow 0^{+}}\left(\ln{\left(\frac{\ln{\left(2\right)}}{\left|\ln{\left(a\right)}\right|}\right)}\right)+\lim _{a\rightarrow 1^{-}}\left(\ln{\left(\frac{\left|\ln{\left(a\right)}\right|}{\ln{\left(2\right)}}\right)}\right)+\lim _{a\rightarrow 1^{+}}\left(\ln{\left(\frac{\ln{\left(3\right)}}{\left|\ln{\left(a\right)}\right|}\right)}\right)$$
- step4: Evaluate the limit:
  $$\left(-\infty\right)+\left(-\infty\right)+\left(+\infty\right)$$
- step5: Calculate:
  $$\textrm{Diverges}$$
Here are the solutions for the integrals provided:

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

73. For the integral 
$$
\int_{1}^{3} \frac{1}{x^{2}} \sqrt{1-\frac{1}{x}} d x,
$$
the result is 
$$
\frac{4\sqrt{6}}{27}.
$$

75. For the integral 
$$
\int_{0}^{3} \frac{d x}{x \ln x},
$$
the integral diverges.

77. For the integral 
$$
\int_{0}^{\pi} \frac{x d x}{1+x^{4}},
$$
the result is 
$$
\frac{\arctan(\pi ^{2})}{2}.
$$

If you have any further questions or need additional assistance, feel free to ask!
Here are the solutions for the integrals: 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. $$ \int_{1}^{3} \frac{1}{x^{2}} \sqrt{1-\frac{1}{x}} d x = \frac{4\sqrt{6}}{27} $$ 75. $$ \int_{0}^{3} \frac{d x}{x \ln x} $$ does not converge. 77. $$ \int_{0}^{\pi} \frac{x d x}{1+x^{4}} = \frac{\arctan(\pi ^{2})}{2} $$

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

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