Question

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

Answer 1

Answer:

$0.8194$

Step-by-step explanation:

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

Answer 2

Answer:

$\displaystyle \int\limits^{\frac{\pi}{2}}_{\frac{\pi}{3}} {(x + \cos x)} \, dx = \frac{5 \pi ^2}{72} + 1 - \frac{\sqrt{3}}{2}$

General Formulas and Concepts:

Calculus

Integration

• Integrals

Integration Rule [Reverse Power Rule]:                                                           $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

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

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

Step-by-step explanation:

Step 1: Define

Identify.

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

Step 2: Integrate

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

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

Unit: Integration