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

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:

$$\int 5x^{3}dx=5\cdot \frac{x^{3+1}}{3+1}+C=\frac{5x^4}{4}+C$$

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:

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

Solution

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

• Differentiate and integrate:

  • Differentiate: $du=dx$

  • Integrate: $v=e^x$

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

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

• 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:

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

Solution:

• Identify the parts: $u=\ln (x)$ and $dv=xdx$

• Differentiate and integrate:

  • Differentiate: $du=\frac{1}{x}dx$

  • Integrate: $v=\frac{x^2}{2}$

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

• Substitute the parts:

$$\int x\ln (x)dx=\ln (x)\cdot \frac{x^2}{2}-\int \frac{x^2}{2}\cdot \frac{1}{x}dx$$

• Simplify the integral:

$$\int x\ln (x)dx=\frac{x^2\ln (x)}{2}-\frac{1}{2}\int xdx$$

• Integrate the remaining part:

$$\frac{1}{2}\int xdx=\frac{1}{2}\cdot \frac{x^2}{2}=\frac{x^2}{4}$$

• Combine the results:

$$\int x\ln (x)dx=\frac{x^2\ln (x)}{2}-\frac{x^2}{4}+C$$

• Final answer:

$$\int x\ln (x)dx=\frac{x^2}{2}\ln (x)-\frac{x^2}{4}+C$$

Advanced Integral

Exercise 6

Function:

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

Solution:

• Identify the substitution: Let $u=x^2$

• Differentiate and solve for $du$:

  • $du=2xdx$

  • $\frac{1}{2}du=xdx$

• Substitute into the integral

$$\int xsin(x^2)dx=\int \sin (u)\cdot \frac{1}{2}du$$

• Integrate:

$$\frac{1}{2}\int \sin (u)du=\frac{1}{2}(-\cos (u))+C$$

• Substitute back into the original variable:

$$\frac{1}{2}(-\cos (u))+C=-\frac{1}{2}\cos (x^2)+C$$

• Final answer

$$\int xsin(x^2)dx=-\frac{1}{2}\cos (x^2)+C$$