Mathematics, 08.01.2021 22:00 573589

Integrate the integral:

$\int \frac{1}{\sqrt{x} \sqrt{1-x} }$

Answers: 3

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Answer from: likevlad2014

$\displaystyle \int {\frac{1}{\sqrt{x} \sqrt{1-x} } } \, dx = 2arcsin(\sqrt{x}) + C$

General Formulas and Concepts:

Calculus

U-SubstitutionIntegration Property: $\int {cf(x)} \, dx = c\int {f(x)} \, dx$ Arctrig Integration: $\displaystyle \int\limits {\frac{1}{\sqrt{a^2-u^2} } } \, du = arcsin(\frac{u}{a} ) + C$

Step-by-step explanation:

Step 1: Define

$\displaystyle \int\limits {\frac{1}{\sqrt{x} \sqrt{1-x} } } \, dx$

Step 2: Identify

Set up variables for u-substitution and integral trig.

$\displaystyleu = \sqrt{x}\\a = 1\\du = \frac{1}{2\sqrt{x}}dx$

Step 3: integrate

Rewrite Integral: $\displaystyle 2\int {\frac{1}{2\sqrt{x} \sqrt{1-x} } } \, dx$ [Integral] U-Substitution: $\displaystyle 2\int {\frac{du}{\sqrt{1^2-(\sqrt{x})^2 } }$ [Integral] U-Sub Arctrig: $\displaystyle 2\int {\frac{du}{\sqrt{1^2-u^2 } }$ [Integral] Arctrig: $\displaystyle 2arcsin(\frac{u}{1}) + C$ Simplify: $\displaystyle 2arcsin(u) + C$ Back-substitute: $\displaystyle 2arcsin(\sqrt{x} ) + C$

Answer from: jhernandezvaldez142

We are given the expression:

$\int\limits {\frac{1}{\sqrt[]{x} }*\frac{1}{\sqrt[]{ 1-x^{2}}} } \, dx$

U-substitution:

let u = $x^{1/2}$ [So, x = u²]

$\frac{du}{dx} = \frac{1}{2\sqrt[]{x}}$ [differentiating both sides wrt x]

dx = du*2√(x)

dx = 2u(du) [Since u = √x]

Finding the Integral:

Plugging these values in the given expression:

$\int\limits {u^{-1}*(1-u^{2})^{-1/2} } \, 2\sqrt[]{x}*du$

$\int\limits {\frac{1}{u}*\frac{1}{\sqrt[]{ 1-u^{2}}}*2u } \, du$

The 'u' in the numerator and denominator will cancel out

$\int\limits {\frac{1}{\sqrt[]{ 1-u^{2}}}*2 } \, du$

Since 2 is a constant

$2\int\limits {\frac{1}{\sqrt[]{ 1-u^{2}}} } \, du$

Using the property: $\int\limits {\frac{1}{\sqrt[]{1-x^{2}} }} \, dx$ = ArcSin(x) + C

2*ArcSin(u) + C

Since u = √x :

2*ArcSin(√x) + C

Answer from: Quest

the answer is 24

step-by-step explanation:

Answer from: Quest

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The diagram represents two statements: p and q which represents regions a, b and c ? i need

The diagram represents two statements: p and q which represents regions a, b and c ? i need

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