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1746. Trigonometric substitutions Evaluate the following integrals. L y4 1 + y2 dy

Calculus: Early Transcendentals | 2nd Edition | ISBN: 9780321947345 | Authors: William L. Briggs ISBN: 9780321947345 167

Solution for problem 36 Chapter 7.4

Calculus: Early Transcendentals | 2nd Edition

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Calculus: Early Transcendentals | 2nd Edition | ISBN: 9780321947345 | Authors: William L. Briggs

Calculus: Early Transcendentals | 2nd Edition

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Problem 36

1746. Trigonometric substitutions Evaluate the following integrals. L y4 1 + y2 dy

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CHM 234 Study Guide 3 __________________________________________________________________ Topic: Conjugated Pi Systems and Pericyclic Reaction Aromatic Compounds Aromatic Substitution Reactions __________________________________________________________________ Conjugated Pi Systems and Pericyclic Reaction Classes of Dienes Cumulated Conjugated Isolated S­trans is more stable than s­cis because of steric hindrance on s­sic Know about LUMO and HOMO Electrophilic Addition It will creates two products: 1,2­adduct 1,4­adduct Thermodynamic Control vs. Kinetic Control 1,2­Adduct 1,4­Adduct Low temperature 0°C 71% 29% High temperature 40°C 15% 85% Introduction to Pericyclic Reactions The three cyclic reaction: Cycloaddition reactions Electrocyclic reaction​ Sigmatropic rearrangements:​ Diels Alder Reaction ● [4+2] cycloaddition Thermodynamic Considerations Conformation of the Diene Reactivity (most to least): ​ 1. s­cis (locked) 2. s­cis 3. s­trans 4. s­trans (locked) The Stereochemistry of the Diels­Alder Reaction: Syn addition If the substituents is cis in the dienophile, then the product will produce a cis product. If the substituents is trnas in the dienophile, then the product will produce a trans product. Endo Rule When the dienophile has a substituent with pi electrons, more of the endo product is formed Endo has a interaction between the developing bond and the electron­withdrawal group Exo has no interaction because it placement is too far away. Pericyclic Reactions D ­ Donating group W ­ Withdrawing group Formation of 1,4 ­ Product Example: Formation of 1,2 ­ Product Example: The formation of 1,3 ­ Product will not form. MO Description of Cycloadditions If there is a conversation of orbital symmetry, then it is ​ symmetry­allowed​ . When the orbital doesn’t overlap, it is ​ symmetry­forbidden.​ When [2+2] cycloaddition is react with heat, the orbital will create ​ symmetry­forbidden​ , so light is the only way a [2+2] cycloaddition can react. Electrocyclic Reactions Thermal Photochemical (light) Four electrons Conrotatory Disrotatory Six electrons Disrotatory Conrotatory Disrotatory​ ­ New sigma orbital will rotate opposite of each other to form a sigma bond. Conrotatory​ ­ New sigma orbital will rotate the same way of each other to form a sigma bond. Six electrons Thermal & Photochemical Four electrons Thermal & Photochemical Sigmatropic Rearrangements The Cope Rearrangement A [3,3] sigmatropic rearrangement where all six of the cyclic system is carbon. The Claisen Rearrangement A [3,3] sigmatropic rearrangement where it is observed to have vinylic ethers. Example: [1,5] sigmatropic rearrangement Example: UV­Vis Spectroscopy The necessary energy, excited electron from HOMO to LUMO, is either the UV or visible region of the spectrum. UV­Vis spectroscopy gives structural information about molecules. Chromophore ​ ­ Absorbing UV­VIS light (The conjugated chain). Auxochromes ​ ­ Group of atoms attached to a chromophore. Aromatic Compounds Nomenclature of Benzene Derivatives Monosubstituted Derivatives of Benzene For some compound, the name comes from the parent name (benzene). Other names: If the substituent chain has a longer carbon chain, then it will become the parent chain. 1­phenylheptane Disubstituted Derivatives of Benzenes Based on the position of two substituents, it will be either ortho, meta, para. Ortho = adjacent [replace 1,2 in naming] Meta = separated by one carbon [replace 1,3 in naming] Para = Opposite of one another [replace 1,4 in naming] Common name for dimethyl benzene is xylene meta­xylene 1,3­dimethylbenzene Polysubstituted Derivatives of Benzenes ● Identify and name the parent ● Identify and name the substituents ● Assign a locant to each substituents ● Arrange the substituents alphabetically Parent chain Name Phenol 4­bromo­2­methylphenol Toluene 5­bromo­2­hydroxytoluene Benzene 4­bromo­1­hydroxy­2­methylbenzene Stability of Benzene Hückel’s Rule A cyclic compound must have odd number of pi electrons pairs to be aromatic. Aromatic Compounds ● Cyclic ● Planer ● Every ring atom must have a p orbital ● Odd number of pair of pi electrons MO Theory and Frost Circles The unpaired electron makes cyclobutadiene very reactive. Antiaromatic Similar to to aromatics’ criteria, but it needs even number of pi electrons pair. Antiaromatic Compounds ● Cyclic ● Planer ● Every ring atom must have a p orbital ● Even number of pair of pi electrons Aromatic compound other than Benzene Nonaromatic Compounds A compound that lacks a continuous systems of p orbitals. Aromatic Antiaromatic 4n+2 4n Example: Aromatic Antiaromatic Antiaromatic Nonaromatic Example: Nonaromatic Aromatic Antiaromatic Aromatic ion Some ring carry formal charge Aromatic Heterocycles Heteroatoms are other atoms that are within the ring other than C or H. Reactions of Benzylic Position A carbon that is attached to a benzene ring is benzylic Oxidation There must be a benzylic position and at least one proton on the Benzylic position. Reagent​ : Na C2 O 2wit7 H SO , 2 O 4 2 Free Radical Bromination Reagent​ : NBS with heat Substitution Reactions of Benzylic Halides (The other way around)↑ Elimination Reactions Hydrogenation Reagent​ : H 2ith Pt Aromatic Substitution Reactions Halogenation Reagent: Br 2with FeBr or3AlBr 3 Cl 2with FeCl or3AlCl 3 (All the mechanisms in this section are the same) Sulfonation Reagent​ : Fuming H SO 2 4 Fuming H​ 2​O​4​consist of sulfuric acid and SO​3​gas. Depending on the concentration of the sulfuric acid: Concentrated = forward reaction Diluted = reverse reaction Nitration Reagent: HNO w3th H SO 2 4 Reduction a Nitro Substituent Reagent​ : H2, Pt Fe, HCl Sn, HCl Friedel­Crafts Alkylation ● Carbon­carbon bonding Reagent: Alkyl halide with Al(halide) 3 As for primary halide, they cannot convert into a carbocation since primary halide are very high in energy. So, they undergo hydride shift. Friedel­Crafts Acylation Clemmensen Reaction Reagent​: Zn(Hg) with HCl and heat Activating Group Substituents that are good electrodonating​ group. Does Ortho​ and Para formation Strong ­ Having lone pairs next to the aromatic ring (Only toward the ring) Moderate ­ Having lone pairs that can delocalize outside the aromatic ring. (Toward the ring and outside the ring) Weak ­ No lone pairs, but inductive effect. (carbon ­ methyl, ethyl, etc.) Deactivation Group Substituents that are good electrowithdrawing​ group. Does Meta​ formation Weak ­ Aromatic with halogen group. (Does ​ortho​ and para formation) Moderate ­ Aromatic ring adjacent to a sp carbon, any double bond or triple bond. Strong ­ Have many withdrawing atom to the adjacent atom. Multiple Substituents Directing effect ● Two group reinforce each other, the new substituent will be located on the position of the reinforced position. [Arrows shows potential placement of the new substituent.] ● Two group oppose each other, the new substituent will be positioned to the more activating group. Steric effect ● The new substituent will be located on the position that will have the least hindrance. If there are two groups, the new substituent will position toward the less steric hindrance group. If there is a directing effect between in the meta position, then it will not position itself in the position because of steric hinderance. Blocking group With the use of sulfonation, the a position can be blocked. Sulfonation can be ​ reversible​ .

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Chapter 7.4, Problem 36 is Solved
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Textbook: Calculus: Early Transcendentals
Edition: 2
Author: William L. Briggs
ISBN: 9780321947345

This full solution covers the following key subjects: . This expansive textbook survival guide covers 128 chapters, and 9720 solutions. The answer to “1746. Trigonometric substitutions Evaluate the following integrals. L y4 1 + y2 dy” is broken down into a number of easy to follow steps, and 13 words. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321947345. The full step-by-step solution to problem: 36 from chapter: 7.4 was answered by , our top Calculus solution expert on 12/23/17, 04:24PM. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 2. Since the solution to 36 from 7.4 chapter was answered, more than 225 students have viewed the full step-by-step answer.

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1746. Trigonometric substitutions Evaluate the following integrals. L y4 1 + y2 dy