# In the poem “Ulysses” by Alfred Lord Tennyson, is Ulysses a heroic or an unheroic figure?

**MATH**

**∫sin(ln(x))xdx∫sin(ln(x))xdx Evaluate the indefinite integral.**

You need to use the following substitution lnx=tlnx=t , such that: lnx=t⇒dxx=dtlnx=t⇒dxx=dt ∫sin(lnx)xdx=∫sintdt∫sin(lnx)xdx=∫sintdt ∫sintdt=−cost+c∫sintdt=-cost+c Replacing back lnxlnx for t…

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**MATH**

**∫etan(x)sec2(x)dx∫etan(x)sec2(x)dx Evaluate the indefinite integral.**

You need to use the following substitution tanx=ttanx=t , such that: tanx=t⇒sec2xdx=dttanx=t⇒sec2xdx=dt ∫etanx⋅sec2xdx=∫etdt=et+c∫etanx⋅sec2xdx=∫etdt=et+cReplacing back tanxtanx for t yields:

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**MATH**

**∫tan−1(x)1+x2dx∫tan-1(x)1+x2dx Evaluate the indefinite integral.**

You need to use the following substitution tan−1x=ttan-1x=t , such that: tan−1x=t⇒dx1+x2=dttan-1x=t⇒dx1+x2=dt ∫(tan−1x)dx1+x2=∫tdt∫(tan-1x)dx1+x2=∫tdt ∫tdt=t22+c∫tdt=t22+c Replacing back…

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**MATH**

**∫5tsin(5t)dt∫5tsin(5t)dt Evaluate the indefinite integral.**

You need to use the following substitution 5t=u5t=u , such that: 5t=u⇒5t⋅ln5dt=du⇒5t⋅dt=duln55t=u⇒5t⋅ln5dt=du⇒5t⋅dt=duln5 ∫5t⋅sin(5t)dt=(1ln5)⋅∫sinudu∫5t⋅sin(5t)dt=(1ln5)⋅∫sinudu

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**MATH**

**∫ecos(t)sin(t)dt∫ecos(t)sin(t)dt Evaluate the indefinite integral.**

You need to use the following substitution cost=ucost=u , such that: cost=u⇒−sintdt=du⇒sintdt=−ducost=u⇒-sintdt=du⇒sintdt=-du ∫ecost⋅sintdt=−∫eudu=−eu+c∫ecost⋅sintdt=-∫eudu=-eu+c Replacing back costcostfor…

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**MATH**

**∫(x2+1)(x3+3x)4dx∫(x2+1)(x3+3x)4dx Evaluate the indefinite integral.**

You need to use the following substitution x3+3x=ux3+3x=u , such that: x3+3x=u⇒(3×2+3)dx=du⇒(x2+1)dx=du3x3+3x=u⇒(3×2+3)dx=du⇒(x2+1)dx=du3 ∫(x2+1)(x3+3x)4dx=(13)⋅∫u4du∫(x2+1)(x3+3x)4dx=(13)⋅∫u4du

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**MATH**

**∫dxax+b,(a≠0)∫dxax+b,(a≠0) Evaluate the indefinite integral.**

∫dxax+b=(1a)⋅∫d(ax+b)ax+b=(1a)⋅ln|ax+b|+C.∫dxax+b=(1a)⋅∫d(ax+b)ax+b=(1a)⋅ln|ax+b|+C. C is an arbitrary constant.

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**MATH**

**∫ex1+ex−−−−−√dx∫ex1+exdx Evaluate the indefinite integral.**

Given ∫ex1+ex−−−−−√dx∫ex1+exdx Integrate using the Substitution Rule. Let u=1+exu=1+ex dudx=exdudx=ex dx=duexdx=duex =∫exu−−√⋅duex=∫exu⋅duex =∫u12du=∫u12du =u3232+C=u3232+C =23(1+ex)32+C=23(1+ex)32+C

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**MATH**

**∫x−−√sin(1+x32)dx∫xsin(1+x32)dx Evaluate the indefinite integral.**

You need to evaluate the indefinite integral by using the substitution 1+x32=t1+x32=t , such that: 1+x32=t⇒(32)x32−1dx=dt⇒x−−√⋅dx=(23)⋅dt1+x32=t⇒(32)x32-1dx=dt⇒x⋅dx=(23)⋅dt

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**MATH**

**∫sec2(θ)tan3(θ)dθ∫sec2(θ)tan3(θ)dθ Evaluate the indefinite integral.**

You need to use the following substitution tanθ=ttanθ=t , such that: tanθ=t⇒(sec2θ)dθ=dttanθ=t⇒(sec2θ)dθ=dt ∫(sec2θ)tan3θdθ=∫t3dt∫(sec2θ)tan3θdθ=∫t3dt ∫t3dt=t44+c∫t3dt=t44+c…

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**MATH**

**∫cos4(θ)sin(θ)dθ∫cos4(θ)sin(θ)dθ Evaluate the indefinite integral.**

You need to use the following substitution cosθ=ucosθ=u , such that: cosθ=u⇒−sinθdθ=du⇒sinθdθ=−ducosθ=u⇒-sinθdθ=du⇒sinθdθ=-du

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**MATH**

**∫(ln(x))2xdx∫(ln(x))2xdx Evaluate the indefinite integral.**

You need to use the following substitution lnx=ulnx=u , such that: lnx=u⇒dxx=dulnx=u⇒dxx=du ∫(ln2x)dxx=∫u2du∫(ln2x)dxx=∫u2du Using the formula ∫undu=un+1n+1+c∫undu=un+1n+1+c yields

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**MATH**

**∫z2z3+1dz∫z2z3+1dz Evaluate the indefinite integral.**

You need to use the following substitution z3+1=uz3+1=u , such that: z3+1=u⇒3z2dz=du⇒z2dz=du3z3+1=u⇒3z2dz=du⇒z2dz=du3 ∫z2⋅dzz3+1=(13)⋅∫1udu∫z2⋅dzz3+1=(13)⋅∫1udu

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**MATH**

**∫a+bx23ax+bx3−−−−−−−−√dx∫a+bx23ax+bx3dx Evaluate the indefinite integral.**

The indefinite integral is (13)⋅∫(d(3ax+bx3)3ax+bx3−−−−−−−−√)=(13)⋅∫(d(3ax+bx3)3ax+bx3)= =(13)⋅2⋅3ax+bx3−−−−−−−−√+C.=(13)⋅2⋅3ax+bx3+C.

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**MATH**

**∫sin(x−−√)x−−√dx∫sin(x)xdx Evaluate the indefinite integral.**

You need to use the following substitution x−−√=ux=u , such that: x−−√=u⇒dx2x−−√=du⇒dxx−−√=2dux=u⇒dx2x=du⇒dxx=2du ∫sin(x−−√)dxx−−√=2⋅∫sinudu∫sin(x)dxx=2⋅∫sinudu

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**MATH**

**∫eu(1−eu)2du∫eu(1-eu)2du Evaluate the indefinite integral.**

You need to use the following substitution 1−eu=t1-eu=t , such that: 1−eu=t⇒−eudu=dt⇒eudu=−dt1-eu=t⇒-eudu=dt⇒eudu=-dt ∫eu⋅du(1−eu)2=−∫dtt2∫eu⋅du(1-eu)2=-∫dtt2 −∫dtt2=1t+c-∫dtt2=1t+c…

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**MATH**

**∫excos(ex)dx∫excos(ex)dx Evaluate the indefinite integral.**

You need to use the following substitution ex=tex=t , such that: ex=t⇒exdx=dtex=t⇒exdx=dt ∫ex⋅cos(ex)dx=∫costdt∫ex⋅cos(ex)dx=∫costdt ∫costdt=sint+c∫costdt=sint+c Replacing back exex for tt yields:…

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**MATH**

**∫sin(πt)dt∫sin(πt)dt Evaluate the indefinite integral.**

You need to use the following substitution π⋅t=uπ⋅t=u , such that: π⋅t=u⇒π⋅dt=du⇒dt=duππ⋅t=u⇒π⋅dt=du⇒dt=duπ ∫sin(π⋅t)dt=(1π)⋅∫sinudu∫sin(π⋅t)dt=(1π)⋅∫sinudu (1π)⋅∫sinudu=−cosu+c(1π)⋅∫sinudu=-cosu+c…

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**MATH**

**∫u1−u2−−−−−√du∫u1-u2du Evaluate the indefinite integral.**

You need to use the following substitution 1−u2=t1-u2=t , such that: 1−u2=t⇒−2udu=dt⇒udu=−dt21-u2=t⇒-2udu=dt⇒udu=-dt2 ∫u⋅1−u2−−−−−√du=−(12)⋅∫t√dt∫u⋅1-u2du=-(12)⋅∫tdt

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**MATH**

**∫dx5−3x∫dx5-3x Evaluate the indefinite integral.**

You need to use the following substitution 5 – 3x = t, such that: 5−3x=t⇒−3dx=dt⇒dx=−dt35-3x=t⇒-3dx=dt⇒dx=-dt3 ∫dx5−3x=−(13)∫dtt∫dx5-3x=-(13)∫dtt −(13)∫dtt=−(13)ln|t|+c-(13)∫dtt=-(13)ln|t|+c…

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**MATH**

**∫sec2(2θ)dθ∫sec2(2θ)dθ Evaluate the indefinite integral.**

You need to use the following substitution 2θ=t2θ=t , such that: 2θ=t⇒2dθ=dt⇒dθ=dt22θ=t⇒2dθ=dt⇒dθ=dt2 ∫sec2(2θ)dθ=(12)⋅∫sec2tdt∫sec2(2θ)dθ=(12)⋅∫sec2tdt

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**MATH**

**∫(x+1)2x+x2−−−−−−√dx∫(x+1)2x+x2dx Evaluate the indefinite integral.**

You need to use the following substitution 2x+x2=t2x+x2=t , such that: 2x+x2=t⇒(2+2x)dx=dt⇒(x+1)dx=dt22x+x2=t⇒(2+2x)dx=dt⇒(x+1)dx=dt2 ∫(x+1)2x+x2−−−−−−√dx=(12)∫t√dt∫(x+1)2x+x2dx=(12)∫tdt

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**MATH**

**∫(3t+2)2.4dt∫(3t+2)2.4dt Evaluate the indefinite integral.**

You need to use the following substitution 3t+2=u3t+2=u , such that: 3t+2=u⇒3dt=du⇒dt=du33t+2=u⇒3dt=du⇒dt=du3 ∫(3t+2)2.4dt=(13)⋅∫u2.4dt∫(3t+2)2.4dt=(13)⋅∫u2.4dt

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**MATH**

**∫(1−2x)9dx∫(1-2x)9dx Evaluate the indefinite integral.**

You need to evaluate the indefinite integral by performing the substitution 1 – 2x = t, such that: 1−2x=t⇒−2dx=dt⇒dx=−dt21-2x=t⇒-2dx=dt⇒dx=-dt2 ∫(1−2x)9dx=−12∫t9dt∫(1-2x)9dx=-12∫t9dt You need to use…

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**MATH**

**∫x2ex3dx∫x2ex3dx Evaluate the indefinite integral.**

You need to evaluate the indefinite integral by performing the substitution x3=tx3=t , such that: x3=t⇒3x2dx=dt⇒x2dx=dt3x3=t⇒3x2dx=dt⇒x2dx=dt3∫x2⋅ex3dx=(13)∫etdt∫x2⋅ex3dx=(13)∫etdt

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**MATH**

**∫xsin(x2)dx∫xsin(x2)dx Evaluate the indefinite integral.**

You need to evaluate the indefinite integral by performing the substitution x2=tx2=t , such that: x2=t⇒2xdx=dt⇒xdx=dt2x2=t⇒2xdx=dt⇒xdx=dt2 ∫x⋅sin(x2)dx=(12)∫sintdt∫x⋅sin(x2)dx=(12)∫sintdt

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**MATH**

**∫sec2(1x)x2dx,u=1x∫sec2(1x)x2dx,u=1x Evaluate the integral by making the given substitution.**

If u=1x,u=1x, then x=1ux=1u and dx=−duu2.dx=-duu2. Therefore the integral is −∫sec2(u)1u2⋅duu2=−∫sec2(u)du=−tan(u)+C=−tan(1x)+C.-∫sec2(u)1u2⋅duu2=-∫sec2(u)du=-tan(u)+C=-tan(1x)+C.

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**MATH**

**∫cos3(θ)sin(θ)dθ,u=cos(θ)∫cos3(θ)sin(θ)dθ,u=cos(θ) Evaluate the integral by making the…**

You need to evaluate the indefinite integral by performing the substitution u=cosθu=cosθ , such that: u=cosθ⇒du=−sinθ⋅dθ⇒sinθ⋅dθ=−duu=cosθ⇒du=-sinθ⋅dθ⇒sinθ⋅dθ=-du

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**MATH**

**∫dt(1−6t)4,u=1−6t∫dt(1-6t)4,u=1-6t Evaluate the integral by making the given substitution.**

Given ∫dt(1−6t)4∫dt(1-6t)4 Integrate using the Substitution Rule. Let u=1−6tu=1-6t dudt=−6dudt=-6 dt=du−6dt=du-6 ∫dtu4∫dtu4 ∫u−4dt∫u-4dt =∫u−4(du−6)=∫u-4(du-6) =−16∫u−4du=-16∫u-4du =−16(u−3−3)+C=-16(u-3-3)+C …

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**MATH**

**∫x2x3+1−−−−−√dx,u=x3+1∫x2x3+1dx,u=x3+1 Evaluate the integral by making the given substitution.**

You need to evaluate the indefinite integral by performing the indicated substitutionu=x3+1u=x3+1 , such that: u=x3+1⇒du=3x2dx⇒x2dx=du3u=x3+1⇒du=3x2dx⇒x2dx=du3

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**MATH**

**∫x3(2+x4)5dx,u=2+x4∫x3(2+x4)5dx,u=2+x4 Evaluate the integral by making the given substitution.**

You need to evaluate the indefinite integral by performing the indicated substitution u=2+x4u=2+x4 , such that: u=2+x4,⇒du=4x3dx⇒x3dx=du4u=2+x4,⇒du=4x3dx⇒x3dx=du4

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**MATH**

**∫e−xdx,u=−x∫e-xdx,u=-x Evaluate the integral by making the given substitution.**

You need to use the following substitution −x=u-x=u , such that: −x=u⇒−dx=du⇒dx=−du-x=u⇒-dx=du⇒dx=-du ∫e−xdx=−∫eudu=−eu+c∫e-xdx=-∫eudu=-eu+c Replacing back −x-x for u yields:

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**SCIENCE**

**What organelles help the nucleus do its job?**

Arguably, all organelles within a cell are dependent upon one another. Each organelle serves a different function that contributes towards the survival of the entire cell. This is the same concept…

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**THE SNIPER**

**What happened to the old woman in “The Sniper” by Liam O’Flaherty?**

A sniper’s bullet hits and kills the old woman from Liam O’Flaherty’s short story “The Sniper.” The old woman in question is actually an informer working for the Free Staters. While the…

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**TO KILL A MOCKINGBIRD**

**In To Kill a Mockingbird, how does finding the gifts in the tree affect Jem?**

Jem is moved that the “haint” of Boo Radley has reached out to them in friendship; he is also impressed with how well Boo has carved the little figures of Scout and himself. When Scout and Jem…

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**OTHELLO**

**Based on Othello’s behavior, is he a victim?**

It’s pretty easy to assume that yes of course Othello is simply the victim of Iago’s machinations. Iago sets out to destroy him with a series of lies, and by the end of the play he has managed to…

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**THE DEVIL AND TOM WALKER**

**What does the narrator suggest might have changed the outcome of the story The Devil and Tom…**

Although the narrator does not directly state a plan of action that would have changed the outcome of the story, there are several significant moments which could have altered Tom’s fate. After…

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**BRAVE NEW WORLD**

**In Brave New World, what prophecy based on consumerism was Huxley implying for our current society?**

One way to summarize the prophecy of the novel: If we do not actively pursue improvements (via science and technology) that are essentially human, we may fall into a trap wherein we instead pursue…

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**THROUGH THE TUNNEL**

**What do the mother and son do every day?**

Each day, Jerry and his mother leave their villa and go to the beach. On the first day of vacation, they go to the same “crowded beach he knew so well from other years.” On that first day, as…

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**THE GREAT GATSBY**

**In The Great Gatsby, Fitzgerald seems to communicate a message about people’s failure to…**

Daisy’s American dream is simultaneously the simplest and the most difficult to achieve of all the main characters in this story, which is perhaps why people seem to gravitate around her. She is…

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**DULCE ET DECORUM EST**

**Based on reading “Dulce et Decorum Est”, do you think the narrator would most likely defend or…**

First, we should distinguish between whether Owen would think a WWIII is likely to happen and whether he would think that England should participate in it. In response to the first question,…

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**JULIUS CAESAR**

**In Julius Caesar, how does Antony change after his funeral oration?**

Brutus sees Antony as non-threatening, as nothing more than Caesar’s right-hand man who can do no harm once Caesar is gone. Cassius thinks they should kill both Antony and Caesar, but Brutus…

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**HAMLET**

**In Hamlet, what does Lord Polonius’ statement, “Your bait of falsehood takes this carp of truth,”…**

Lord Polonius says this to Reynaldo, his servant, in Act 2 scene 1. He is sending Rynaldo to Paris to give him some notes and money, but more importantly, to spy on his son. Laertes has been given…

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**SCIENCE**

Magma is generated in the area of the lower part of the earth’s crust and the upper part of its hot center, the mantle. It consists of rock that has been liquified by heat, and is a mixture made up…

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**INTO THE WILD**

**I think Chris McCandless was a hero in Into the Wild. What points from the book should I use to…**

When writing an argument in favor of McCandless, be prepared to meet a crossfire of opinions regarding the topic. Therefore, to argue that McCandless is a “hero”, you should have both the evidence…

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**A SEPARATE PEACE**

**In A Separate Peace by John Knowles, does Gene ever go to the war?**

Gene Forrester, the main character in A Separate Peace, does eventually enlist in the war, but he never leaves the country or sees battle. The story focuses on the summer before his senior year,…

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**LONDON**

**How does the speaker view London in William Blake’s poem of the same name?**

The first stanza already expresses a negative tone, for the speaker notes that as he wanders through the mapped streets of London, he perceives on every face that he meets, ‘Marks of weakness,…

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**ULYSSES**

**In the poem “Ulysses” by Alfred Lord Tennyson, is Ulysses a heroic or an unheroic figure?**

To answer this question, you must decide what you think is “heroic” and why. Ulysses presents himself, at the beginning of the poem, as a rather pathetic figure. He is an “idle king,” living among…

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**TO KILL A MOCKINGBIRD**

**Where does Calpurnia take the children?**

While Atticus is away for work in law, Calpurnia is in charge of the children. When Sunday comes, she wants to take them to church. So, she along with Jem and Scout go to Calpurnia’s church. It…

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**SCIENCE**

**Carbon-12 and carbon-13 are isotopes of carbon. What does this mean?**

Hello! Each chemical element (and carbon among them) is defined by the number of protons in its nucleus. Also the same number of electrons are found in each atom of the same element. For carbon,…

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