In Problems 21-26, find each indefinite integral using the given substitution. 21. $$ \int e^{3 x+1} d x ; $$ let $$ u=3 x+1 $$ 22. $$ \int \frac{d x}{x \ln x} ; $$ let $$ u=\ln x $$ 23. $$ \int\left(1-t^{2}\right)^{6} t d t $$; let $$ u=\left(1-t^{2}\right) $$ 438 81. 24. $$ \int \sin ^{5} x \cos x d x $$; let $$ u=\sin x $$ 25. $$ \int \frac{x^{2} d x}{\sqrt{1-x^{6}}} ; $$ let $$ u=x^{3} $$ 26. $$ \int \frac{e^{-x}}{6+e^{-x}} d x $$; let $$ u=6+e^{-x} $$

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Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \frac{1}{u} du$$
- step1: Evaluate the integral:
  $$\ln{\left(\left|u\right|\right)}$$
- step2: Add the constant of integral C:
  $$\ln{\left(\left|u\right|\right)} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int e^{u} \frac{1}{3} du $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int e^{u}\times \frac{1}{3} du$$
- step1: Multiply the terms:
  $$\int \frac{e^{u}}{3} du$$
- step2: Rewrite the expression:
  $$\int \frac{1}{3}e^{u} du$$
- step3: Use properties of integrals:
  $$\frac{1}{3}\times \int e^{u} du$$
- step4: Evaluate the integral:
  $$\frac{1}{3}e^{u}$$
- step5: Multiply the terms:
  $$\frac{e^{u}}{3}$$
- step6: Add the constant of integral C:
  $$\frac{e^{u}}{3} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int u^{6} (-\frac{1}{2}) du $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int u^{6}\left(-\frac{1}{2}\right) du$$
- step1: Reorder the terms:
  $$\int -\frac{1}{2}u^{6} du$$
- step2: Use properties of integrals:
  $$-\frac{1}{2}\times \int u^{6} du$$
- step3: Evaluate the integral:
  $$-\frac{1}{2}\times \frac{u^{6+1}}{6+1}$$
- step4: Simplify:
  $$-\frac{1}{2}\times \frac{u^{7}}{7}$$
- step5: Multiply the terms:
  $$-\frac{u^{7}}{2\times 7}$$
- step6: Multiply the terms:
  $$-\frac{u^{7}}{14}$$
- step7: Add the constant of integral C:
  $$-\frac{u^{7}}{14} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{1}{u} du $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \frac{1}{u} du$$
- step1: Evaluate the integral:
  $$\ln{\left(\left|u\right|\right)}$$
- step2: Add the constant of integral C:
  $$\ln{\left(\left|u\right|\right)} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int u^{5} du $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int u^{5} du$$
- step1: Evaluate the integral:
  $$\frac{u^{5+1}}{5+1}$$
- step2: Add the numbers:
  $$\frac{u^{6}}{5+1}$$
- step3: Add the numbers:
  $$\frac{u^{6}}{6}$$
- step4: Add the constant of integral C:
  $$\frac{u^{6}}{6} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{u^{2}}{\sqrt{1-u^{2}}} \frac{1}{3} du $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \frac{u^{2}}{\sqrt{1-u^{2}}}\times \frac{1}{3} du$$
- step1: Multiply the terms:
  $$\int \frac{u^{2}}{3\sqrt{1-u^{2}}} du$$
- step2: Rewrite the expression:
  $$\int \frac{1}{3}\times \frac{u^{2}}{\sqrt{1-u^{2}}} du$$
- step3: Use properties of integrals:
  $$\frac{1}{3}\times \int \frac{u^{2}}{\sqrt{1-u^{2}}} du$$
- step4: Evaluate the integral:
  $$\frac{1}{3}\left(\frac{1^{2}}{2}\arcsin\left(u\right)-\frac{1}{2}u\sqrt{1^{2}-u^{2}}\right)$$
- step5: Simplify:
  $$\frac{1}{3}\left(\frac{1}{2}\arcsin\left(u\right)-\frac{1}{2}u\sqrt{1-u^{2}}\right)$$
- step6: Use the distributive property:
  $$\frac{1}{3}\times \frac{1}{2}\arcsin\left(u\right)+\frac{1}{3}\left(-\frac{1}{2}u\sqrt{1-u^{2}}\right)$$
- step7: Multiply the terms:
  $$\frac{1}{6}\arcsin\left(u\right)+\frac{1}{3}\left(-\frac{1}{2}u\sqrt{1-u^{2}}\right)$$
- step8: Multiply the terms:
  $$\frac{1}{6}\arcsin\left(u\right)-\frac{1}{6}u\sqrt{1-u^{2}}$$
- step9: Add the constant of integral C:
  $$\frac{1}{6}\arcsin\left(u\right)-\frac{1}{6}u\sqrt{1-u^{2}} + C, C \in \mathbb{R}$$
Let's solve each of the integrals step by step using the given substitutions.

### 21. $$ \int e^{3x+1} dx $$; let $$ u = 3x + 1 $$

Using the substitution $$ u = 3x + 1 $$, we find $$ du = 3dx $$ or $$ dx = \frac{du}{3} $$.

Now, substituting into the integral:
$$
\int e^{3x+1} dx = \int e^{u} \frac{du}{3} = \frac{1}{3} e^{u} + C = \frac{1}{3} e^{3x+1} + C
$$

### 22. $$ \int \frac{dx}{x \ln x} $$; let $$ u = \ln x $$

Using the substitution $$ u = \ln x $$, we find $$ du = \frac{1}{x} dx $$ or $$ dx = x du = e^{u} du $$.

Now, substituting into the integral:
$$
\int \frac{dx}{x \ln x} = \int \frac{1}{u} du = \ln |u| + C = \ln |\ln x| + C
$$

### 23. $$ \int (1-t^{2})^{6} t dt $$; let $$ u = (1-t^{2}) $$

Using the substitution $$ u = 1 - t^{2} $$, we find $$ du = -2t dt $$ or $$ dt = -\frac{du}{2t} $$.

Now, substituting into the integral:
$$
\int (1-t^{2})^{6} t dt = -\frac{1}{2} \int u^{6} du = -\frac{1}{2} \cdot \frac{u^{7}}{7} + C = -\frac{u^{7}}{14} + C = -\frac{(1-t^{2})^{7}}{14} + C
$$

### 24. $$ \int \sin^{5} x \cos x dx $$; let $$ u = \sin x $$

Using the substitution $$ u = \sin x $$, we find $$ du = \cos x dx $$.

Now, substituting into the integral:
$$
\int \sin^{5} x \cos x dx = \int u^{5} du = \frac{u^{6}}{6} + C = \frac{\sin^{6} x}{6} + C
$$

### 25. $$ \int \frac{x^{2} dx}{\sqrt{1-x^{6}}} $$; let $$ u = x^{3} $$

Using the substitution $$ u = x^{3} $$, we find $$ du = 3x^{2} dx $$ or $$ dx = \frac{du}{3x^{2}} $$.

Now, substituting into the integral:
$$
\int \frac{x^{2} dx}{\sqrt{1-x^{6}}} = \int \frac{1}{\sqrt{1-u^{2}}} \frac{du}{3} = \frac{1}{3} \arcsin(u) + C = \frac{1}{3} \arcsin(x^{3}) + C
$$

### 26. $$ \int \frac{e^{-x}}{6+e^{-x}} dx $$; let $$ u = 6 + e^{-x} $$

Using the substitution $$ u = 6 + e^{-x} $$, we find $$ du = -e^{-x} dx $$ or $$ dx = -\frac{du}{e^{-x}} = -\frac{du}{u - 6} $$.

Now, substituting into the integral:
$$
\int \frac{e^{-x}}{6+e^{-x}} dx = -\int \frac{1}{u} du = -\ln |u| + C = -\ln |6 + e^{-x}| + C
$$

### Summary of Results
1. $$ \frac{1}{3} e^{3x+1} + C $$
2. $$ \ln |\ln x| + C $$
3. $$ -\frac{(1-t^{2})^{7}}{14} + C $$
4. $$ \frac{\sin^{6} x}{6} + C $$
5. $$ \frac{1}{3} \arcsin(x^{3}) + C $$
6. $$ -\ln |6 + e^{-x}| + C $$
Here are the solutions to the integrals: 1. $$ \frac{1}{3} e^{3x+1} + C $$ 2. $$ \ln |\ln x| + C $$ 3. $$ -\frac{(1-t^{2})^{7}}{14} + C $$ 4. $$ \frac{\sin^{6} x}{6} + C $$ 5. $$ \frac{1}{3} \arcsin(x^{3}) + C $$ 6. $$ -\ln |6 + e^{-x}| + C $$

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