sites.math.rutgers.edufq15/244S14/Memo1-1.pdf · . . . . . . Logarithm Functions De nitions: f(x) =...

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...... Review of Formulas in Calculus

Fei Qi

Rutgers University

fq15@math.rutgers.edu

January 21, 2014

Fei Qi (Rutgers University) Review of Formulas in Calculus January 21, 2014 1 / 16

. . . . . .

Disclaimer

The slides are written exclusively for 244 students. It might not beappropriate to use them in any earlier course.

There may be errors. Use them at your own discretion. Anyone whonotify me with an error will get some award in grade points.

. . . . . .

Power Function

Definitions: f (x) = xa (a ∈ R)

Derivative:

f ′(x) = (xa)′ =

{axa−1 a ̸= 0

0 a = 0

Antiderivative:∫f (x)dx =

∫xadx =

a+ 1xa+1 + C a ̸= −1

ln |x |+ C a = −1

Note: When you perform the integration, you should never forget totake absolute values. However in many cases of the 244 course, youdon’t have to care too much about that.

. . . . . .

Power Function

Definitions: f (x) = xa (a ∈ R)Derivative:

f ′(x) = (xa)′ =

{axa−1 a ̸= 0

0 a = 0

∫xadx =

a+ 1xa+1 + C a ̸= −1

ln |x |+ C a = −1

. . . . . .

Power Function

f ′(x) = (xa)′ =

{axa−1 a ̸= 0

0 a = 0

∫xadx =

a+ 1xa+1 + C a ̸= −1

ln |x |+ C a = −1

. . . . . .

Power Function

f ′(x) = (xa)′ =

{axa−1 a ̸= 0

0 a = 0

∫xadx =

a+ 1xa+1 + C a ̸= −1

ln |x |+ C a = −1

. . . . . .

Exponential Functions

Definitions: f (x) = ax (a > 0)

Derivative:f ′(x) = (ax)′ = ax ln a.

How to compute: Take logarithms on both sides and apply thedifferentiation law of composite functions.

Antiderivative: ∫f (x)dx =

∫axdx =

ln aax + C

How to compute: Make use of the derivative above.

. . . . . .

∫axdx =

ln aax + C

. . . . . .

∫axdx =

ln aax + C

. . . . . .

∫axdx =

ln aax + C

. . . . . .

∫axdx =

ln aax + C

. . . . . .

Logarithm Functions

Definitions: f (x) = loga x (a > 0)

Derivative:

f ′(x) = (loga x)′ =

x ln a.

How to compute: Strictly speaking you should be using the law forinverse functions. But if you know already that (ln x)′ = 1/x , thenyou can simply make use of the fact loga x = ln x/ ln a.

∫loga xdx =

ln a(x ln x − x) + C

How to compute: Use integration by parts to solve the special casethat a = e, then again use loga x = ln x/ ln a.

. . . . . .

Logarithm Functions

Derivative:

f ′(x) = (loga x)′ =

x ln a.

∫loga xdx =

. . . . . .

Logarithm Functions

Derivative:

f ′(x) = (loga x)′ =

x ln a.

∫loga xdx =

. . . . . .

Logarithm Functions

Derivative:

f ′(x) = (loga x)′ =

x ln a.

∫loga xdx =

. . . . . .

Logarithm Functions

Derivative:

f ′(x) = (loga x)′ =

x ln a.

∫loga xdx =

. . . . . .

Trigonometric functions

Definitions: sin x , cos x , tan x , cot x , sec x , csc x

Derivative:

(sin x)′ = cos x , (cos x)′ = − sin x ,

(tan x)′ = sec2 x , (cot x)′ = − csc2 x ,

(sec x)′ = sec x tan x , (csc x)′ = − csc x cot x .

How to compute: Use definitions of derivatives and the trigonometricidentities to work on sin x and cos x . Use laws of quotients to work ontan x and cot x . Use either law of quotients or chain rule to work onsec x and csc x .

. . . . . .

Derivative:

. . . . . .

Derivative:

. . . . . .

Derivative:

. . . . . .

Antiderivative:∫sin xdx = − cos x + C ,

∫cos xdx = sin x + C ,∫

tan xdx = − ln | cos x |+ C ,

∫cot xdx = ln | sin x |+ C ,∫

sec xdx = ln | sec x + tan x |+ C ,

∫csc x = − ln | csc x + cot x |.

How to compute: Use the derivatives above to see the first two.Write in quotients and use substitutions then you will see the secondtwo. Use trigonometric techniques to get the last two.

. . . . . .

Antiderivative:∫sin xdx = − cos x + C ,

∫cos xdx = sin x + C ,∫

tan xdx = − ln | cos x |+ C ,

∫cot xdx = ln | sin x |+ C ,∫

sec xdx = ln | sec x + tan x |+ C ,

∫csc x = − ln | csc x + cot x |.

How to compute: Use the derivatives above to see the first two.Write in quotients and use substitutions then you will see the secondtwo. Use trigonometric techniques to get the last two.

. . . . . .

Details of∫sec xdx

∫sec xdx =

cos xdx

∫cos x

cos2 xdx

1− sin2 xd sin x =

1− sin x+

1 + sin x

)d sin x

2(ln |1 + sin x | − ln |1− sin x |) =

∣∣∣∣1 + sin x

1− sin x

∣∣∣∣It would be fine to end here. This is a correct answer. It just take a fewmore steps to get what we are looking for

. . . . . .

∫sec xdx =

cos xdx =

∫cos x

cos2 xdx

1− sin x+

1 + sin x

)d sin x

2(ln |1 + sin x | − ln |1− sin x |) =

∣∣∣∣1 + sin x

1− sin x

. . . . . .

∫sec xdx =

cos xdx =

∫cos x

cos2 xdx

1− sin2 xd sin x

1− sin x+

1 + sin x

)d sin x

2(ln |1 + sin x | − ln |1− sin x |) =

∣∣∣∣1 + sin x

1− sin x

. . . . . .

∫sec xdx =

cos xdx =

∫cos x

cos2 xdx

1− sin x+

1 + sin x

)d sin x

2(ln |1 + sin x | − ln |1− sin x |) =

∣∣∣∣1 + sin x

1− sin x

. . . . . .

∫sec xdx =

cos xdx =

∫cos x

cos2 xdx

1− sin x+

1 + sin x

)d sin x

2(ln |1 + sin x | − ln |1− sin x |)

∣∣∣∣1 + sin x

1− sin x

. . . . . .

∫sec xdx =

cos xdx =

∫cos x

cos2 xdx

1− sin x+

1 + sin x

)d sin x

2(ln |1 + sin x | − ln |1− sin x |) =

∣∣∣∣1 + sin x

1− sin x

∣∣∣∣

It would be fine to end here. This is a correct answer. It just take a fewmore steps to get what we are looking for

. . . . . .

∫sec xdx =

cos xdx =

∫cos x

cos2 xdx

1− sin x+

1 + sin x

)d sin x

2(ln |1 + sin x | − ln |1− sin x |) =

∣∣∣∣1 + sin x

1− sin x

. . . . . .

∫sec xdx =

cos xdx =

∫cos x

cos2 xdx

1− sin x+

1 + sin x

)d sin x

2(ln |1 + sin x | − ln |1− sin x |) = 1

∣∣∣∣1 + sin x

1− sin x

∣∣∣∣

∣∣∣∣(1 + sin x)(1 + sin x)

(1− sin x)(1 + sin x)

∣∣∣∣ =1

∣∣∣∣(1 + sin x)2

cos2 x

∣∣∣∣= ln

∣∣∣∣1 + sin x

∣∣∣∣ = ln |sec x + tan x |

. . . . . .

∫sec xdx =

cos xdx =

∫cos x

cos2 xdx

1− sin x+

1 + sin x

)d sin x

2(ln |1 + sin x | − ln |1− sin x |) = 1

∣∣∣∣1 + sin x

1− sin x

∣∣∣∣=

∣∣∣∣(1 + sin x)(1 + sin x)

(1− sin x)(1 + sin x)

∣∣∣∣

∣∣∣∣(1 + sin x)2

cos2 x

∣∣∣∣= ln

∣∣∣∣1 + sin x

∣∣∣∣ = ln |sec x + tan x |

. . . . . .

∫sec xdx =

cos xdx =

∫cos x

cos2 xdx

1− sin x+

1 + sin x

)d sin x

2(ln |1 + sin x | − ln |1− sin x |) = 1

∣∣∣∣1 + sin x

1− sin x

∣∣∣∣=

∣∣∣∣(1 + sin x)(1 + sin x)

(1− sin x)(1 + sin x)

∣∣∣∣ =1

∣∣∣∣(1 + sin x)2

cos2 x

∣∣∣∣

∣∣∣∣1 + sin x

∣∣∣∣ = ln |sec x + tan x |

. . . . . .

∫sec xdx =

cos xdx =

∫cos x

cos2 xdx

1− sin x+

1 + sin x

)d sin x

2(ln |1 + sin x | − ln |1− sin x |) = 1

∣∣∣∣1 + sin x

1− sin x

∣∣∣∣=

∣∣∣∣(1 + sin x)(1 + sin x)

(1− sin x)(1 + sin x)

∣∣∣∣ =1

∣∣∣∣(1 + sin x)2

cos2 x

∣∣∣∣= ln

∣∣∣∣1 + sin x

∣∣∣∣ = ln |sec x + tan x |

. . . . . .

Inverse Trigonometric functions

Definitions: arcsin x , arccos x , arctan x , arccotx .

Derivative:

(arcsin x)′ =1√

1− x2, (arccos x)′ = − 1√

1− x2,

(arctan x)′ =1

1 + x2, (arccotx)′ = − 1

1 + x2.

How to compute: Use the techniques dealing with inverse functions.Example of computing arctan x :

y = arctan x ⇒ x = tan y

⇒ dx = sec2 ydy = (1 + tan2 y)dy = (1 + x2)dy ⇒ dy

1 + x2.

Antiderivative: Not interesting at least in 244. So forget it.

. . . . . .

Derivative:

(arcsin x)′ =1√

1− x2, (arccos x)′ = − 1√

1− x2,

(arctan x)′ =1

1 + x2, (arccotx)′ = − 1

1 + x2.

. . . . . .

Derivative:

(arcsin x)′ =1√

1− x2, (arccos x)′ = − 1√

1− x2,

(arctan x)′ =1

1 + x2, (arccotx)′ = − 1

1 + x2.

How to compute: Use the techniques dealing with inverse functions.

Example of computing arctan x :

1 + x2.

. . . . . .

Derivative:

(arcsin x)′ =1√

1− x2, (arccos x)′ = − 1√

1− x2,

(arctan x)′ =1

1 + x2, (arccotx)′ = − 1

1 + x2.

. . . . . .

Derivative:

(arcsin x)′ =1√

1− x2, (arccos x)′ = − 1√

1− x2,

(arctan x)′ =1

1 + x2, (arccotx)′ = − 1

1 + x2.

⇒ dx = sec2 ydy = (1 + tan2 y)dy = (1 + x2)dy

⇒ dy

1 + x2.

. . . . . .

Derivative:

(arcsin x)′ =1√

1− x2, (arccos x)′ = − 1√

1− x2,

(arctan x)′ =1

1 + x2, (arccotx)′ = − 1

1 + x2.

. . . . . .

Derivative:

(arcsin x)′ =1√

1− x2, (arccos x)′ = − 1√

1− x2,

(arctan x)′ =1

1 + x2, (arccotx)′ = − 1

1 + x2.

. . . . . .

Hyperbolic trigonometric functions

Definitions:

sinh x =ex − e−x

2, cosh x =

ex + e−x

2, tanh x =

sinh x

cosh x, coth x =

cosh x

sinh x.

Derivative:(sinh x)′ = cosh x , (cosh x)′ = sinh x .

The rest two are left as exercises for product rule.

How to compute: Straightforward.

Antiderivative:∫sinh xdx = − cosh x + C ,

∫cosh xdx = sinh x + C .

The rest two are left as exercises for technique of substitution.

. . . . . .

Definitions:

2, cosh x =

ex + e−x

2, tanh x =

sinh x

cosh x, coth x =

cosh x

sinh x.

. . . . . .

Definitions:

2, cosh x =

ex + e−x

2, tanh x =

sinh x

cosh x, coth x =

cosh x

sinh x.

. . . . . .

Definitions:

2, cosh x =

ex + e−x

2, tanh x =

sinh x

cosh x, coth x =

cosh x

sinh x.

. . . . . .

Definitions:

2, cosh x =

ex + e−x

2, tanh x =

sinh x

cosh x, coth x =

cosh x

sinh x.

. . . . . .

Definitions:

2, cosh x =

ex + e−x

2, tanh x =

sinh x

cosh x, coth x =

cosh x

sinh x.

. . . . . .

More formulas

a2 + x2=

aarctan

How to compute: Substitution by scalar.

x2 − a2=

∣∣∣∣a− x

∣∣∣∣+ C

How to compute: Either by trigonometric substitution or by breakingrational functions.

. . . . . .

More formulas

a2 + x2=

aarctan

x2 − a2=

∣∣∣∣a− x

∣∣∣∣+ C

. . . . . .

More formulas

a2 + x2=

aarctan

x2 − a2=

∣∣∣∣a− x

∣∣∣∣+ C

. . . . . .

More formulas

a2 + x2=

aarctan

x2 − a2=

∣∣∣∣a− x

∣∣∣∣+ C

. . . . . .

More formulas

∫1√

a2 − x2= arcsin

How to compute: Again substitution by scalar.

∫1√

x2 ± a2= ln |x +

√x2 ± a2|+ C

How to compute: Either by trigonometric substitution or byhyperbolic substitution.

. . . . . .

More formulas

∫1√

a2 − x2= arcsin

∫1√

x2 ± a2= ln |x +

√x2 ± a2|+ C

. . . . . .

More formulas

∫1√

a2 − x2= arcsin

∫1√

x2 ± a2= ln |x +

√x2 ± a2|+ C

. . . . . .

More formulas

∫1√

a2 − x2= arcsin

∫1√

x2 ± a2= ln |x +

√x2 ± a2|+ C

. . . . . .

More detail about the last integral

Example: ∫1√

x2 − a2

Approach by trigonometric substitution: Let x = a sec t.∫1√

x2 − a2dx =

a√sec2t − 1

tan t· sin t

cos2tdt =

cos tdt =

∫d sin t

1− sin2t= ln

∣∣∣∣1 + sin t

1− sin t

∣∣∣∣= ln

∣∣∣∣∣(1 + sin t)2

∣∣∣∣∣ = ln

∣∣∣∣1 + sin t

∣∣∣∣ = ln | sec t + tan t|

∣∣∣∣∣∣xa +x√

1− (a/x)2

∣∣∣∣∣∣+ C = ln |x +√

x2 − a2|+ C

. . . . . .

Example: ∫1√

x2 − a2

Approach by trigonometric substitution: Let x = a sec t.

∫1√

x2 − a2dx =

a√sec2t − 1

tan t· sin t

cos2tdt =

cos tdt =

∫d sin t

1− sin2t= ln

∣∣∣∣1 + sin t

1− sin t

∣∣∣∣= ln

∣∣∣∣∣(1 + sin t)2

∣∣∣∣∣ = ln

∣∣∣∣1 + sin t

∣∣∣∣ = ln | sec t + tan t|

∣∣∣∣∣∣xa +x√

1− (a/x)2

∣∣∣∣∣∣+ C = ln |x +√

x2 − a2|+ C

. . . . . .

Example: ∫1√

x2 − a2

x2 − a2dx

a√sec2t − 1

tan t· sin t

cos2tdt =

cos tdt =

∫d sin t

1− sin2t= ln

∣∣∣∣1 + sin t

1− sin t

∣∣∣∣= ln

∣∣∣∣∣(1 + sin t)2

∣∣∣∣∣ = ln

∣∣∣∣1 + sin t

∣∣∣∣ = ln | sec t + tan t|

∣∣∣∣∣∣xa +x√

1− (a/x)2

∣∣∣∣∣∣+ C = ln |x +√

x2 − a2|+ C

. . . . . .

Example: ∫1√

x2 − a2

x2 − a2dx =

a√sec2t − 1

tan t· sin t

cos2tdt =

cos tdt =

∫d sin t

1− sin2t= ln

∣∣∣∣1 + sin t

1− sin t

∣∣∣∣= ln

∣∣∣∣∣(1 + sin t)2

∣∣∣∣∣ = ln

∣∣∣∣1 + sin t

∣∣∣∣ = ln | sec t + tan t|

∣∣∣∣∣∣xa +x√

1− (a/x)2

∣∣∣∣∣∣+ C = ln |x +√

x2 − a2|+ C

. . . . . .

Example: ∫1√

x2 − a2

x2 − a2dx =

a√sec2t − 1

tan t· sin t

cos2tdt

cos tdt =

∫d sin t

1− sin2t= ln

∣∣∣∣1 + sin t

1− sin t

∣∣∣∣= ln

∣∣∣∣∣(1 + sin t)2

∣∣∣∣∣ = ln

∣∣∣∣1 + sin t

∣∣∣∣ = ln | sec t + tan t|

∣∣∣∣∣∣xa +x√

1− (a/x)2

∣∣∣∣∣∣+ C = ln |x +√

x2 − a2|+ C

. . . . . .

Example: ∫1√

x2 − a2

x2 − a2dx =

a√sec2t − 1

tan t· sin t

cos2tdt =

cos tdt

∫d sin t

1− sin2t= ln

∣∣∣∣1 + sin t

1− sin t

∣∣∣∣= ln

∣∣∣∣∣(1 + sin t)2

∣∣∣∣∣ = ln

∣∣∣∣1 + sin t

∣∣∣∣ = ln | sec t + tan t|

∣∣∣∣∣∣xa +x√

1− (a/x)2

∣∣∣∣∣∣+ C = ln |x +√

x2 − a2|+ C

. . . . . .

Example: ∫1√

x2 − a2

x2 − a2dx =

a√sec2t − 1

tan t· sin t

cos2tdt =

cos tdt =

∫d sin t

1− sin2t=

∣∣∣∣1 + sin t

1− sin t

∣∣∣∣= ln

∣∣∣∣∣(1 + sin t)2

∣∣∣∣∣ = ln

∣∣∣∣1 + sin t

∣∣∣∣ = ln | sec t + tan t|

∣∣∣∣∣∣xa +x√

1− (a/x)2

∣∣∣∣∣∣+ C = ln |x +√

x2 − a2|+ C

. . . . . .

Example: ∫1√

x2 − a2

x2 − a2dx =

a√sec2t − 1

tan t· sin t

cos2tdt =

cos tdt =

∫d sin t

1− sin2t= ln

∣∣∣∣1 + sin t

1− sin t

∣∣∣∣

∣∣∣∣∣(1 + sin t)2

∣∣∣∣∣ = ln

∣∣∣∣1 + sin t

∣∣∣∣ = ln | sec t + tan t|

∣∣∣∣∣∣xa +x√

1− (a/x)2

∣∣∣∣∣∣+ C = ln |x +√

x2 − a2|+ C

. . . . . .

Example: ∫1√

x2 − a2

x2 − a2dx =

a√sec2t − 1

tan t· sin t

cos2tdt =

cos tdt =

∫d sin t

1− sin2t= ln

∣∣∣∣1 + sin t

1− sin t

∣∣∣∣= ln

∣∣∣∣∣(1 + sin t)2

∣∣∣∣∣

∣∣∣∣1 + sin t

∣∣∣∣ = ln | sec t + tan t|

∣∣∣∣∣∣xa +x√

1− (a/x)2

∣∣∣∣∣∣+ C = ln |x +√

x2 − a2|+ C

. . . . . .

Example: ∫1√

x2 − a2

x2 − a2dx =

a√sec2t − 1

tan t· sin t

cos2tdt =

cos tdt =

∫d sin t

1− sin2t= ln

∣∣∣∣1 + sin t

1− sin t

∣∣∣∣= ln

∣∣∣∣∣(1 + sin t)2

∣∣∣∣∣ = ln

∣∣∣∣1 + sin t

∣∣∣∣

= ln | sec t + tan t|

∣∣∣∣∣∣xa +x√

1− (a/x)2

∣∣∣∣∣∣+ C = ln |x +√

x2 − a2|+ C

. . . . . .

Example: ∫1√

x2 − a2

x2 − a2dx =

a√sec2t − 1

tan t· sin t

cos2tdt =

cos tdt =

∫d sin t

1− sin2t= ln

∣∣∣∣1 + sin t

1− sin t

∣∣∣∣= ln

∣∣∣∣∣(1 + sin t)2

∣∣∣∣∣ = ln

∣∣∣∣1 + sin t

∣∣∣∣ = ln | sec t + tan t|

∣∣∣∣∣∣xa +x√

1− (a/x)2

∣∣∣∣∣∣+ C = ln |x +√

x2 − a2|+ C

. . . . . .

Example: ∫1√

x2 − a2

x2 − a2dx =

a√sec2t − 1

tan t· sin t

cos2tdt =

cos tdt =

∫d sin t

1− sin2t= ln

∣∣∣∣1 + sin t

1− sin t

∣∣∣∣= ln

∣∣∣∣∣(1 + sin t)2

∣∣∣∣∣ = ln

∣∣∣∣1 + sin t

∣∣∣∣ = ln | sec t + tan t|

∣∣∣∣∣∣xa +x√

1− (a/x)2

∣∣∣∣∣∣+ C

= ln |x +√

x2 − a2|+ C

. . . . . .

Example: ∫1√

x2 − a2

x2 − a2dx =

a√sec2t − 1

tan t· sin t

cos2tdt =

cos tdt =

∫d sin t

1− sin2t= ln

∣∣∣∣1 + sin t

1− sin t

∣∣∣∣= ln

∣∣∣∣∣(1 + sin t)2

∣∣∣∣∣ = ln

∣∣∣∣1 + sin t

∣∣∣∣ = ln | sec t + tan t|

∣∣∣∣∣∣xa +x√

1− (a/x)2

∣∣∣∣∣∣+ C = ln |x +√

x2 − a2|+ C

. . . . . .

Example: ∫1√

x2 + a2

Approach by hyperbolic substitution: Let x = a sinh t.

∫1√

x2 + a2dx =

sinh2t + 1d(a sinh t)

a cosh ta cosh tdt =

∫dt = t + C

x = a sinh t =et − e−t

2⇒ 2

aet = e2t − 1 ⇒ e2t − 2

aet − 1 = 0

⇒ et =x +

√x2 + a2

a(The smaller root makes et negative)∫

1√x2 + a2

dx = t + C = ln

∣∣∣∣∣x ±√x2 + a2

∣∣∣∣∣+ C

. . . . . .

Example: ∫1√

x2 + a2

Approach by hyperbolic substitution: Let x = a sinh t.∫1√

x2 + a2dx

∫dt = t + C

2⇒ 2

aet = e2t − 1 ⇒ e2t − 2

aet − 1 = 0

⇒ et =x +

√x2 + a2

1√x2 + a2

dx = t + C = ln

∣∣∣∣∣x ±√x2 + a2

∣∣∣∣∣+ C

. . . . . .

Example: ∫1√

x2 + a2

x2 + a2dx =

∫dt = t + C

2⇒ 2

aet = e2t − 1 ⇒ e2t − 2

aet − 1 = 0

⇒ et =x +

√x2 + a2

1√x2 + a2

dx = t + C = ln

∣∣∣∣∣x ±√x2 + a2

∣∣∣∣∣+ C

. . . . . .

Example: ∫1√

x2 + a2

x2 + a2dx =

a cosh ta cosh tdt

∫dt = t + C

2⇒ 2

aet = e2t − 1 ⇒ e2t − 2

aet − 1 = 0

⇒ et =x +

√x2 + a2

1√x2 + a2

dx = t + C = ln

∣∣∣∣∣x ±√x2 + a2

∣∣∣∣∣+ C

. . . . . .

Example: ∫1√

x2 + a2

x2 + a2dx =

∫dt = t + C

2⇒ 2

aet = e2t − 1 ⇒ e2t − 2

aet − 1 = 0

⇒ et =x +

√x2 + a2

1√x2 + a2

dx = t + C = ln

∣∣∣∣∣x ±√x2 + a2

∣∣∣∣∣+ C

. . . . . .

Example: ∫1√

x2 + a2

x2 + a2dx =

∫dt = t + C

⇒ 2x

aet = e2t − 1 ⇒ e2t − 2

aet − 1 = 0

⇒ et =x +

√x2 + a2

1√x2 + a2

dx = t + C = ln

∣∣∣∣∣x ±√x2 + a2

∣∣∣∣∣+ C

. . . . . .

Example: ∫1√

x2 + a2

x2 + a2dx =

∫dt = t + C

2⇒ 2

aet = e2t − 1

⇒ e2t − 2x

aet − 1 = 0

⇒ et =x +

√x2 + a2

1√x2 + a2

dx = t + C = ln

∣∣∣∣∣x ±√x2 + a2

∣∣∣∣∣+ C

. . . . . .

Example: ∫1√

x2 + a2

x2 + a2dx =

∫dt = t + C

2⇒ 2

aet = e2t − 1 ⇒ e2t − 2

aet − 1 = 0

⇒ et =x +

√x2 + a2

1√x2 + a2

dx = t + C = ln

∣∣∣∣∣x ±√x2 + a2

∣∣∣∣∣+ C

. . . . . .

Example: ∫1√

x2 + a2

x2 + a2dx =

∫dt = t + C

2⇒ 2

aet = e2t − 1 ⇒ e2t − 2

aet − 1 = 0

⇒ et =x +

√x2 + a2

a(The smaller root makes et negative)

∫1√

x2 + a2dx = t + C = ln

∣∣∣∣∣x ±√x2 + a2

∣∣∣∣∣+ C

. . . . . .

Example: ∫1√

x2 + a2

x2 + a2dx =

∫dt = t + C

2⇒ 2

aet = e2t − 1 ⇒ e2t − 2

aet − 1 = 0

⇒ et =x +

√x2 + a2

1√x2 + a2

dx = t + C = ln

∣∣∣∣∣x ±√x2 + a2

∣∣∣∣∣+ C

. . . . . .

The End

sites.math.rutgers.edufq15/244S14/Memo1-1.pdf · . . . . . . Logarithm Functions De nitions: f(x) =...

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