Answer:
[tex]\displaystyle \int {\big[ 7\theta - 6csc(\theta)cot(\theta) \big]} \, d\theta = 6csc(\theta) + \frac{7x^2}{2} + C[/tex]
General Formulas and Concepts:
Calculus
Integration
- Integrals
- [Indefinite Integrals] Integration Constant C
Integration Rule [Reverse Power Rule]: [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Property [Multiplied Constant]: [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
Integration Property [Addition/Subtraction]: [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle \int {\big[ 7\theta - 6csc(\theta)cot(\theta) \big]} \, d\theta[/tex]
Step 2: Integrate
- [Integral] Rewrite [Integration Property - Addition/Subtraction]: [tex]\displaystyle \int {\big[ 7\theta - 6csc(\theta)cot(\theta) \big]} \, d\theta = \int {7\theta} \, d\theta - \int {6csc(\theta)cot(\theta)} \, d\theta[/tex]
- [Integrals] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle \int {\big[ 7\theta - 6csc(\theta)cot(\theta) \big]} \, d\theta = 7\int {\theta} \, d\theta - 6\int {csc(\theta)cot(\theta)} \, d\theta[/tex]
- [1st Integral] Reverse Power Rule: [tex]\displaystyle \int {\big[ 7\theta - 6csc(\theta)cot(\theta) \big]} \, d\theta = 7 \Big( \frac{\theta^2}{2} \Big) - 6\int {csc(\theta)cot(\theta)} \, d\theta[/tex]
- [Integral] Trigonometric Integration: [tex]\displaystyle \int {\big[ 7\theta - 6csc(\theta)cot(\theta) \big]} \, d\theta = 7 \Big( \frac{\theta^2}{2} \Big) - 6[-csc(\theta)] + C[/tex]
- Simplify: [tex]\displaystyle \int {\big[ 7\theta - 6csc(\theta)cot(\theta) \big]} \, d\theta = 6csc(\theta) + \frac{7x^2}{2} + C[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
