Math Problem Statement
Solution
The integral provided is:
The\ problem[ involves\ evaluatingint the_ definite0 integral^{\sqrt{3}} :
frac{4x}{[ \sqrtint{x_^02^{ +\ sqrt1}} , dx ]
Step-by{-Step3 Solution}}:
1. Substitution:
Let u = x^ \2frac + {41x,}{\ sosqrt that{x: ^\2[ +du = 21x}} \ ,, dx dx. \]
quad ###text Step-by{or-Step Solution} \quad:
1 x. \ ,Sub dxstitution = : frac { Let1 (}{ u2 = x} du^ 2\ +]
Also1, \ update). the Then, limits of ( integration du:
- = When 2 xx \ =, dx0 \,), and ( the u integral = becomes: 0 ^2 + \1[ = \int_0^{\sqrt{3}} \frac{ 41x }{)
- When ( x = \sqrt{\3}sqrt {x), (^ u2 = + (\ sqrt1{}}3 }),^ dx2 = + \ int1_{ =u (30 +)} ^{u(\sqrt{3})} \frac{4}{\1 =sqrt {4u }})
####cd 2ot. Rewrite the integral: Substituting \ into thefrac integral, we{ get: 1}{2[ }\int _,0 du^{. \ \sqrt]
{32}}. \ Simplfracify: {4x }{\ The limitssqrt{x of^ integration2 change + as follows1: }} \ -, When dx ( x = = \int _01 ^),4 ( \ ufrac ={ 40}{^\2sqrt +{ u}} \cdot \frac{1}{2} \1, = du 1\ ),
- When]
, ( the u constants =: (\sqrt[ {\3})int^_12^4 + 1 = \4frac{4}{ \).
sqrt{u }} \cd The integral becomes:
2}{\sqrt{ u \}}] \3,. du ** =Evaluate**: The ant2ider \ivative ofint_ \(1 u^^{-14/ u^{-1/22} \}) is \ \( , du2$$ u^{####1/2 3}. \ Evaluate). the So integral: : The \ integral[ of \( u2^{- \1int/_21^} \4) u is^{-: 1$$/\2int} u \, du^{-1 =/ 2 \cdot \2left}[ \, du = 22u^{u1^{/12/} 2\}$$ rightApply the]_ limits1 of^ integration4: =\ [ 2 \4 \intleft[_ u1^^{4 u^{-1/12/2} \, du = 2 \left}[ \ right2]_u1^{^14/. 2 \] Substitute} the limits: \ \[ 4 \rightleft]_[1 \^sqrt{4 =4 }4 - \ \leftsqrt[{ u1^{}1/ \2right] = 4 \cdot (2 -} \1right)]_ =1 ^44. \ \] ] Sub###stitute Final the Answer limits: : \\[ [ \4int \_left0[^{ u\^{sqrt1{/32}}} \ \fracright{]_41x^}{4\ =sqrt {x4^ \2left +[ \1sqrt}}{ \4,} dx - = \ sqrt4{. 1\}] \Wouldright you] like = a deeper4 explanation \ ofleft any[ step or2 additional - examples ?1 \--- right###] Follow =-Up Questions4: \1cd.ot What is1 the = substitution rule4 in integration\ and] how### does Final it Answer simplify: problems\? [ 2\.boxed How{ do4 you} determine\ new] limitsLet of me integration know after if substitution you'd? like3 more. details Why or is further the clar antificationsider!ivative of--- \(### u Exp^{-anding1 Learning/: 2Here} are \ )5 equal related to questions \( you might explore2: u1^{.1 How/ do you decide2 on a} substitution when solving \ integr)? als4? .2 How. What are can some other substitution examples be of used in definite integr moreals complex involving integr squareals roots? 5? . How do definite integr3. Canals this differ integral be solved using from a indefinite different integr methodals,? like--- tr###igon Tipometric: substitutionAlways? verify4. your What is the significance substitution by differenti ofating the it limits back of to check if integration it in matches definite the integrals original? integral's5 structure.! How does substitution simplify complex integrals? ### Tip: When performing substitution in definite integrals, always remember to change the limits of integration to match the new variable. This avoids reverting back to the original variable unnecessarily.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Definite Integrals
Substitution
Trigonometric Substitution
Integration Techniques
Formulas
∫(from a to b) f(x) dx
Substitution formula: u = g(x), du = g'(x) dx
Theorems
Fundamental Theorem of Calculus
Substitution Method for Integrals
Suitable Grade Level
Grades 11-12