AE 14: Integration

In this exercise, we will:

• Practice Integration:

  • Apply the basic integration rules to find the integrals of simple polynomial functions.

• Apply Integration by Parts and Substitution:

  • Use these techniques to integrate more complex functions involving products and compositions.

• Solve Advanced Integration Problems:

  • Tackle integrals of functions with nested compositions and multiple variables, reinforcing the use of integration by parts and substitution.

Common functions + their Integrals

1. Power Function:

• Function:

$$f(x)=x^n$$

• Integral:

$$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C(n\ne -1)$$

2. Exponential Function:

• Function:

$$f(x)=e^x$$

• Integral:

$$\int e^{x}dx=e^x+C$$

• Function:

$$f(x)=a^x$$

• Integral:

$$\int a^{x}dx=\frac{a^x}{\ln (a)}+C$$

3. Natural Logarithm

• Function:

$$f(x)=ln(x)$$

• Integral:

$$\int \ln (x)dx=x\ln (x)-x+C$$

4. Trigonometric Functions

• Function:

$$f(x)=\sin (x)$$

• Integral:

$$\int \sin (x)dx=-\cos (x)+C$$

• Function:

$$f(x)=\cos (x)$$

• Integral:

$$\int \cos (x)dx=\sin (x)+C$$

• Function:

$$f(x)=\tan (x)$$

• Integral:

$$\int \tan (x)dx=-\ln |\cos (x)|+C=\ln |\sec (x)|+C$$

5. Hyperbolic Functions

• Function:

$$f(x)=\sinh (x)$$

• Integral:

$$\int \sinh (x)dx=\cosh (x)+C$$

• Function:

$$f(x)=\cosh (x)$$

• Integral:

$$\int \cosh (x)dx=\sinh (x)+C$$

Integrals

For each problem, find the integral of the given function. Show all steps clearly.

Exercise 1

Function:

$$f(x)=5x^3$$

Solution:

• The power rule for integration states that $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$

• Applying the power rule:

add response here.

Exercise 2

Function:

$$g(x)=\sqrt{x}$$

Solution:

• Rewrite the function with a fractional exponent: add response here.

• Apply the power rule:

add response here.

Exercise 3

Function:

$$h(x)=\ln (x)$$

Solution:

• Use the integral of the natural logarithm function:

add response here.

Intermediate Derivatives Using Chain Rule and Product Rule

Exercise 4

Function:

$$\int x{e}^{x}dx$$

Solution

• Identify the parts: Let $u=x$ and $dv=e^{x}dx$

• Differentiate and integrate:

  • Differentiate: add response here.

  • Integrate: add response here.

• Apply the integration by parts formula: $\int udv=uv-\int vdu$

add response here.

• Simplify the integral:

add response here.

• Final answer:

add response here.

Exercise 5

Function:

$$\int x\ln (x)dx$$

Solution:

• Identify the parts: ___ and ___

• Differentiate and integrate:

  • Differentiate: add response here.

  • Integrate: add response here.

• Apply the integration by parts formula: $\int udv=uv-\int vdu$

• Substitute the parts:

add response here.

• Simplify the integral:

add response here.

• Integrate the remaining part:

add response here.

• Combine the results:

add response here.

• Final answer:

add response here.

Advanced Integral

Exercise 6

Function:

$$\int x\sin (x^2)dx$$

Solution:

• Identify the substitution: Let ___

• Differentiate and solve for ___:

  • add response here.

  • add response here.

• Substitute into the integral

add response here.

• Integrate:

add response here.

• Substitute back into the original variable:

add response here.

• Final answer

add response here.