AE 14: Integration
Suggested answers
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:
\[ \int 5x^3 \, dx = 5 \cdot \frac{x^{3+1}}{3+1} + C = \frac{5x^4}{4} + C \]
Exercise 2
Function:
\[ g(x) = \sqrt{x} \]
Solution:
Rewrite the function with a fractional exponent: \(\sqrt{x}=x^{1/2}\).
Apply the power rule:
\[ g(x)=x^{1/2} \]
\[ \int x^{1/2} \, dx = \int x^{1/2} \, dx = \frac{x^{1/2+1}}{1/2+1} + C = \frac{x^{3/2}}{3/2} + C = \frac{2}{3} x^{3/2} + C \]
Exercise 3
Function:
\[ h(x)=\ln(x) \]
Solution:
- Use the integral of the natural logarithm function:
\[ \int \ln(x) \, dx = x \ln(x) - x + C \]
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: \(du = dx\)
Integrate: \(v = e^x\)
Apply the integration by parts formula: \(\int u \space dv = uv - \int v \space du\)
\[ \int x e^x \, dx = x e^x - \int e^x \, dx \]
- Simplify the integral:
\[ \int x e^x \, dx = x e^x - e^x + C \]
- Final answer:
\[ \int x e^x \, dx = e^x (x - 1) + C \]
Exercise 5
Function:
\[ \int x \ln(x) \space dx \]
Solution:
Identify the parts: \(u=\ln(x)\) and \(dv=x \space dx\)
Differentiate and integrate:
Differentiate: \(du=\frac{1}{x}\space dx\)
Integrate: \(v=\frac{x^{2}}{2}\)
Apply the integration by parts formula: \(\int u \space dv = uv - \int v \space du\)
Substitute the parts:
\[ \int x \space \ln(x) \space dx = \ln(x) \cdot \frac{x^2}{2} - \int \frac{x^2}{2} \cdot \frac{1}{x} \space dx \]
- Simplify the integral:
\[ \int x \ln(x) \space dx = \frac{x^{2} \ln(x)}{2} - \frac{1}{2} \int x \space dx \]
- Integrate the remaining part:
\[ \frac{1}{2} \int x \space dx = \frac{1}{2} \cdot \frac{x^2}{2} = \frac{x^2}{4} \]
- Combine the results:
\[ \int x \ln(x) \space dx = \frac{x^2 \ln(x)}{2} - \frac{x^2}{4} + C \]
- Final answer:
\[ \int x \ln(x) \space dx = \frac{x^2}{2} \ln(x) - \frac{x^2}{4} + C \]
Advanced Integral
Exercise 6
Function:
\[ \int x \sin (x^2) \space dx \]
Solution:
Identify the substitution: Let \(u = x^2\)
Differentiate and solve for \(du\):
\(du = 2x \space dx\)
\(\frac{1}{2}du = x \space dx\)
Substitute into the integral
\[ \int x sin(x^2) \space dx = \int \sin(u) \cdot \frac{1}{2} \space du \]
- Integrate:
\[ \frac{1}{2} \int \sin(u) \space du = \frac{1}{2}(-\cos(u)) + C \]
- Substitute back into the original variable:
\[ \frac{1}{2}(-\cos(u)) + C = -\frac{1}{2}\cos(x^2) + C \]
- Final answer
\[ \int x sin(x^2) \space dx = -\frac{1}{2}\cos(x^2) + C \]