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Integrate the function $\frac{-10x^2+5x+5}{2\sqrt{1-x-x^2}}$ from $0.618$ to 0

Step-by-step Solution

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Solution

Final answer to the problem

$\frac{25}{8}\arcsin\left(\frac{2\left(0.618+\frac{1}{2}\right)}{\sqrt{5}}\right)-\frac{5}{2}\cdot \left(0.618+\frac{1}{2}\right)\sqrt{- \left(0.618+\frac{1}{2}\right)^2+\frac{5}{4}}+\frac{5}{4}\arcsin\left(\frac{0.618+\frac{1}{2}}{\sqrt{5}}\right)+\frac{15}{2}\sqrt{- \left(0.618+\frac{1}{2}\right)^2+\frac{5}{4}}+\frac{5}{4}\arcsin\left(\frac{2\left(0.618+\frac{1}{2}\right)}{\sqrt{5}}\right)-\frac{5}{2}\arcsin\left(\frac{2\left(0.618+\frac{1}{2}\right)}{\sqrt{5}}\right)-\left(\frac{25}{8}\arcsin\left(\frac{2\left(0+\frac{1}{2}\right)}{\sqrt{5}}\right)-\frac{5}{2}\cdot \left(0+\frac{1}{2}\right)\sqrt{- \left(0+\frac{1}{2}\right)^2+\frac{5}{4}}+\frac{5}{4}\arcsin\left(\frac{0+\frac{1}{2}}{\sqrt{5}}\right)+\frac{15}{2}\sqrt{- \left(0+\frac{1}{2}\right)^2+\frac{5}{4}}+\frac{5}{4}\arcsin\left(\frac{2\left(0+\frac{1}{2}\right)}{\sqrt{5}}\right)-\frac{5}{2}\arcsin\left(\frac{2\left(0+\frac{1}{2}\right)}{\sqrt{5}}\right)\right)$
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Step-by-step Solution

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Since the upper limit of the integral is less than the lower one, we can rewrite the limits by applying the inverse property of integration limits: If we invert the limits of an integral, it changes sign: $\int_a^bf(x)dx=-\int_b^af(x)dx$

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$-\int_{0}^{0.618}\frac{-10x^2+5x+5}{2\sqrt{1-x-x^2}}dx$

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Learn how to solve problems step by step online. Integrate the function (-10.0x^2+5x+5)/(2(1-x-x^2)^(1/2)) from 0.618 to 0. Since the upper limit of the integral is less than the lower one, we can rewrite the limits by applying the inverse property of integration limits: If we invert the limits of an integral, it changes sign: \int_a^bf(x)dx=-\int_b^af(x)dx. Take the constant \frac{1}{2} out of the integral. Multiply the fraction and term in - \left(\frac{1}{2}\right)\int\frac{-10x^2+5x+5}{\sqrt{1-x-x^2}}dx. Rewrite the expression \frac{-10x^2+5x+5}{\sqrt{1-x-x^2}} inside the integral in factored form.

Final answer to the problem

$\frac{25}{8}\arcsin\left(\frac{2\left(0.618+\frac{1}{2}\right)}{\sqrt{5}}\right)-\frac{5}{2}\cdot \left(0.618+\frac{1}{2}\right)\sqrt{- \left(0.618+\frac{1}{2}\right)^2+\frac{5}{4}}+\frac{5}{4}\arcsin\left(\frac{0.618+\frac{1}{2}}{\sqrt{5}}\right)+\frac{15}{2}\sqrt{- \left(0.618+\frac{1}{2}\right)^2+\frac{5}{4}}+\frac{5}{4}\arcsin\left(\frac{2\left(0.618+\frac{1}{2}\right)}{\sqrt{5}}\right)-\frac{5}{2}\arcsin\left(\frac{2\left(0.618+\frac{1}{2}\right)}{\sqrt{5}}\right)-\left(\frac{25}{8}\arcsin\left(\frac{2\left(0+\frac{1}{2}\right)}{\sqrt{5}}\right)-\frac{5}{2}\cdot \left(0+\frac{1}{2}\right)\sqrt{- \left(0+\frac{1}{2}\right)^2+\frac{5}{4}}+\frac{5}{4}\arcsin\left(\frac{0+\frac{1}{2}}{\sqrt{5}}\right)+\frac{15}{2}\sqrt{- \left(0+\frac{1}{2}\right)^2+\frac{5}{4}}+\frac{5}{4}\arcsin\left(\frac{2\left(0+\frac{1}{2}\right)}{\sqrt{5}}\right)-\frac{5}{2}\arcsin\left(\frac{2\left(0+\frac{1}{2}\right)}{\sqrt{5}}\right)\right)$

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Function Plot

Plotting: $\frac{-10x^2+5x+5}{2\sqrt{1-x-x^2}}$

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x
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(◻)
+
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×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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