Question

Evaluate the definite integral from pi/3 to pi/2 of (x+cosx) dx

Answer 1

`0.8194`

Explanation:

`\int\limits^(\pi)/(2) _(\pi)/(3) {x+cos(x)} \, dx\\\\=(1)/(2)x^2+sin(x)\Bigr|_{(\pi)/(3)}^{(\pi)/(2)}\\\\=[(1)/(2)((\pi)/(2))^2+sin((\pi)/(2))]-[(1)/(2)((\pi)/(3))^2+sin((\pi)/(3))]\\\\=[(1)/(2)((\pi^2)/(4))+1]-[(1)/(2)((\pi^2)/(9))+(√(3))/(2)]\\ \\ =(\pi^2)/(8)-(\pi^2)/(18)+1-(√(3))/(2)\\ \\ \approx0.8194`

Answer 2

`\displaystyle \int\limits^{(\pi)/(2)}_{(\pi)/(3)} {(x + \cos x)} \, dx = (5 \pi ^2)/(72) + 1 - (√(3))/(2)`

General Formulas and Concepts:

Calculus

Integration

• Integrals

Integration Rule [Reverse Power Rule]:
`\displaystyle \int {x^n} \, dx = (x^(n + 1))/(n + 1) + C`

Integration Rule [Fundamental Theorem of Calculus 1]:
`\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)`

Integration Property [Addition/Subtraction]:
`\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx`

Explanation:

Step 1: Define

Identify.

`\displaystyle \int\limits^{(\pi)/(2)}_{(\pi)/(3)} {(x + \cos x)} \, dx`

Step 2: Integrate

1. [Integral] Rewrite [Integration Property - Addition/Subtraction]:
   `\displaystyle \int\limits^{(\pi)/(2)}_{(\pi)/(3)} {(x + \cos x)} \, dx = \int\limits^{(\pi)/(2)}_{(\pi)/(3)} {x} \, dx + \int\limits^{(\pi)/(2)}_{(\pi)/(3)} {\cos x} \, dx`
2. [Left Integral] Integration Rule [Reverse Power Rule]:
   `\displaystyle \int\limits^{(\pi)/(2)}_{(\pi)/(3)} {(x + \cos x)} \, dx = (x^2)/(2) \bigg| \limits^{(\pi)/(2)}_{(\pi)/(3)} + \int\limits^{(\pi)/(2)}_{(\pi)/(3)} {\cos x} \, dx`
3. [Right Integral] Trigonometric Integration:
   `\displaystyle \int\limits^{(\pi)/(2)}_{(\pi)/(3)} {(x + \cos x)} \, dx = (x^2)/(2) \bigg| \limits^{(\pi)/(2)}_{(\pi)/(3)} + \sin x \bigg| \limits^{(\pi)/(2)}_{(\pi)/(3)}`
4. Integration Rule [Fundamental Theorem of Calculus 1]:
   `\displaystyle \int\limits^{(\pi)/(2)}_{(\pi)/(3)} {(x + \cos x)} \, dx = (5 \pi ^2)/(72) + \bigg( 1 - (√(3))/(2) \bigg)`

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration