Honors Calculus II

Review 3 - Some of the Answers

Question 1

Integrate[(Sin[x])^5, {x, 0, Pi/2}]

Integrate[(Sin[2  x])^4 (Cos[2  x])^4, x]

Integrate[(Sin[x])^6 (Cos[x])^2, x]

Integrate[(Tan[x])^3 (Sec[x])^3, x]

Integrate[(Tan[x] + Cot[x])^2, x]

Integrate[(Sec[x])^4, x]

Integrate[(Cos[x])^3 (Sin[x])^2, x]

Integrate[(Tan[x])^6, {x, 0, Pi/4}]

Question 2

Integrate[x/Sqrt[4 - x^2], x]

Integrate[x^2/Sqrt[9 - x^2], x]

Integrate[1/(x^2*Sqrt[x^2 - 16]),  {x, 5, 8}]

Integrate[1/Sqrt[x^2 + 2  x + 5], x]

Integrate[Sqrt[5 - 4  x - x^2], x]

Integrate[1/(16 + x^2)^(3/2), x]

Integrate[Sqrt[4 - x^2]/x^2, x]

Integrate[Sqrt[x^2 - 4]/x, {x, 2, 4}]

Question 3

Limit[(Sin[x] - 2  x)/x,  x -> 0]

Limit[(ArcSin[x])/(3*ArcTan[x]), x -> 0]

Limit[(8^Sqrt[x] - 1)/(3^Sqrt[x] - 1), x -> 0]

Limit[(2*Sqrt[x + 1] - x - 2)/(2  x^2), x -> 0]

Limit[(Exp[x] - Exp[-x])/(4*Sin[x]), x -> 0]

Limit[Log[Sin[x]]/(Pi/2 - x), x -> Pi/2]

Limit[(Sqrt[x] - x)/Log[x], x -> 1]

Limit[(Pi/2 - x) Tan[x], x -> Pi/2]

Question 4

Integrate[x/Sqrt[9 + x^2], {x, 3, Infinity}]

Integrate[1/(x*Log[x]), {x, 2, Infinity}]

Integrate[1/(x^2 + 2  x + 5), {x, -Infinity, Infinity}]

Integrate[x*Exp[-x], {x, 0, Infinity}]

Question 5

Integrate[1/Sqrt[1 - x^2], {x, 0, 1}]

Integrate[1/Sqrt[9 - x], {x, 0, 9}]

Integrate[Tan[2  x], {x, 0, Pi/4}]

Integrate[Csc[x], {x, 0, Pi/2}]

Integrate[Log[x]/x,  {x, 0, 1}]

Integrate[1/(4 - x^2), {x, -2, 2}]

Integrate[x/(x^2 - 4)^(2/3), {x, -3, 0}]

Integrate[1/(x + 1)^(4/3), {x, -2, -1}]

Question 6

Integrate[x/Log[x], {x, 2, Infinity}]

NIntegrate[Sqrt[x + 1]/x^2, {x, 1, Infinity}]

NIntegrate[1/Sqrt[x + x^4], {x, 1, Infinity}]

NIntegrate[1/(Exp[x] - 2^x), {x, 1, Infinity}]

Question 7

This is the midpoint sum for 50 subintervals  corresponding to the integral of Cos[x] over the interval [0,10]

NIntegrate[Cos[x], {x, 0, 10}]

Question 8

Integrate[(x^(3/5) + x^(1/6))/Sqrt[x], x]

Integrate[1/(Sqrt[x - 1] + Sqrt[x + 1]), x]

Integrate[(Exp[x] + Exp[2  x] + Exp[3  x])/Exp[4  x], x]

Integrate[(Tan[x])^2, x]

Integrate[(8  x^2 + 6  x + 4)/(x + 1), x]

Integrate[Exp[x]*Sin[Exp[x]], x]

Integrate[Log[x]/x, x]

Integrate[Exp[Exp[x]]*Exp[x], x]

Integrate[Log[Cos[x]]*Tan[x], x]

Integrate[Log[Log[x]]/(x*Log[x]), x]

Integrate[x^2 (1 + x^3)^(10), x]

You probably have the answer in a better form than this

Integrate[(1 + Sqrt[x])^5/Sqrt[x], x]

Integrate[x*(Sin[x^2])^3  Cos[x^2], x]

Integrate[(Sec[x] - Tan[x])^2, x]

Integrate[1/(1 + Sin[x]), x]

Integrate[x*Sqrt[2 - x^2], x]

Integrate[x*Sqrt[2 - x], x]

Integrate[x*(x^2 + 1)^3, x]

Integrate[1/(2  x^2 - 3  x + 2), {x, 1, 2}]

Integrate[Sec[x] Tan[x], {x, -Pi/3, Pi/3}]

Question 11

D[Log[Cosh[3  x]], x]

D[ArcCosh[Sinh[x]], x]

D[x*ArcTanh[x], x]

Question 12

Integrate[4*Exp[-x]*Sinh[x], x]

Integrate[Tanh[x], {x, 0, Log[2]}]

Integrate[Cosh[Sqrt[x]]/Sqrt[x], x]