Integration following Multiple Derivations

Calculus Level 3

Using the following formulas: { f ( x ) = 5 cos ⁡ 2 ( x ) + sin ⁡ 2 ( x ) e x g ( x ) = f ( x ) − f ′ ( x ) h ( x ) = f ( x ) f ′ ( x ) g ( x ) g ′ ( x ) \begin{cases}f(x) = \dfrac{5^{\cos ^2(x)+\sin ^2(x)}}{e^x} \\ g(x) = f(x) - f'(x) \\ h(x) = \dfrac{f(x)f'(x)}{g(x)g'(x)}\end{cases}

Determine: ∫ 0 100 h ( x ) d x \displaystyle \int_{0}^{100} h(x) dx

50 100 25 0 75

View wiki
David Hontz by David Hontz , 26, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

David Hontz
May 15, 2016

f ( x ) = 5 c o s 2 ( x ) + s i n 2 ( x ) e x = 5 e x f ′ ( x ) = − 5 e − x = − 5 e x g ( x ) = 5 e x − − 5 e x = 5 + 5 e x = 10 e x g ′ ( x ) = − 10 e x h ( x ) = 5 e x − 5 e x 10 e x − 10 e x = − 25 − 100 = 1 4 ∫ 0 100 h ( x ) d x = ∫ 0 100 1 4 d x = 1 4 ( 100 − 0 ) = 25 f(x) =\frac{5^{cos^2(x)+sin^2(x)}}{e^x} = \boxed{\frac{5}{e^x}} \\ f'(x) = -5e^{-x} = \boxed{\frac{-5}{e^x}} \\ g(x) = \frac{5}{e^x} - \frac{-5}{e^x} = \frac{5+5}{e^x} = \boxed{\frac{10}{e^x}} \\ g'(x) = \boxed{\frac{-10}{e^x}} \\ h(x) = \frac{\frac{5}{e^x} \frac{-5}{e^x}}{\frac{10}{e^x} \frac{-10}{e^x}} = \frac{-25}{-100} = \boxed{\frac{1}{4}} \\ \int_{0}^{100} h(x) dx = \int_{0}^{100} \frac{1}{4} dx = \frac{1}{4} (100-0) = \boxed{25}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...