Lesson 21.1 - Table of Laplace Transforms

Table Title: Common Laplace Transforms

tsL[f(t)]F(s)L[eat]1s−aL[tn]n!sn+1L[sin⁡(bt)]bs2+b2L[cos⁡(bt)]ss2+b2L[eatf(t)]F(s−a)L[eattn]n!(s−a)n+1L[eatsin⁡(bt)]b(s−a)2+b2L[eatcos⁡(bt)]s−a(s−a)2+b2

This table can be used for reference. Rather than using the definition of Laplace transforms, you can memorize this table or reference it to take the Laplace transform of a function. We start with the function on the left side in the t domain and transform it to the function on the right in the s domain.

Let's do some practice.

Example

  1. L[t2−3t−2e−tsin⁡(3t)]

Recall the property of linearity. This means that we can take the Laplace transform of these functions individually.

L[t2−3t−2e−tsin⁡(3t)]=L[t2]−3L[t]−2L[e−tsin⁡(3t)]

Referencing the table:

L[t2]=2!s3=2⋅1s3=2s3

−3L[t]=−3⋅1s2=−3s2

−2L[e−tsin⁡(3t)]=−2⋅2(s−(−1))2+22=−4(s+1)2+4

Combining this all together:

L[t2−3t−2e−tsin⁡(3t)]=2s3−3s2−4(s+1)2+4

Try for Yourself

  1. L[t4−t2−4t+sin⁡(2t)]
  2. L[4sin⁡(2t)]+t2−e2t]

Trig Functions

It's worth mentioning that there will often be times where you cannot directly apply a Laplace transform to a function. In these situations, we may need to identify some trig identity in order to transform the function. Let's recall some important trig identities:

  1. sin⁡tcos⁡t=12sin⁡(2t)
  2. sin2⁡t=12(1−cos⁡(2t))
  3. cos2⁡t=12(1+cos⁡(2t))
  4. sin⁡Acos⁡B=12[sin⁡(A+B)+sin⁡(A−B)]
  5. sin⁡Asin⁡B=12[cos⁡(A−B)−cos⁡(A+B)]
  6. cos⁡Acos⁡B=12[cos⁡(A−B)+cos⁡(A−B)]

Next Lesson: Lesson 22 - Inverse Laplace Transforms

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