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
- Power Function:
- Function:
\[ f(x)=x^n \]
- Integral:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) \]
- Exponential Function:
- Function:
\[ f(x)=e^x \]
- Integral:
\[ \int e^x \, dx = e^x + C \]
- Function:
\[ f(x)=a^x \]
- Integral:
\[ \int a^x \, dx = \frac{a^x}{\ln(a)} + C \]
- Natural Logarithm
- Function:
\[ f(x) = ln(x) \]
- Integral:
\[ \int \ln(x) \, dx = x \ln(x) - x + C \]
- 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:
\[ \int \tan(x) \, dx = -\ln|\cos(x)| + C = \ln|\sec(x)| + C \]
- 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 \(\int x^n \space dx = \frac{x^{n+1}}{n+1}+C\)
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:
\[ \int xe^x \space dx \]
Solution
Identify the parts: Let \(u=x\) and \(dv=e^x \space dx\)
Differentiate and integrate:
Differentiate: add response here.
Integrate: add response here.
Apply the integration by parts formula: \(\int u \space dv = uv - \int v \space du\)
add response here.
- Simplify the integral:
add response here.
- Final answer:
add response here.
Exercise 5
Function:
\[ \int x \ln(x) \space dx \]
Solution:
Identify the parts: ___ and ___
Differentiate and integrate:
Differentiate: add response here.
Integrate: add response here.
Apply the integration by parts formula: \(\int u \space dv = uv - \int v \space du\)
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:
\[ \int x \sin (x^2) \space dx \]
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.