Evaluate the integrals \int e^{t}\cos (3e^{t}-2)dt

sputavanomr

sputavanomr

Answered question

2021-11-23

Evaluate the integrals etcos(3et2)dt

Answer & Explanation

Nicole Keller

Nicole Keller

Beginner2021-11-24Added 14 answers

Step 1
We have to evaluate the integral:
etcos(3et2)dt
In this case we should use substitution method since derivatives of one function is in the integral.
So let x=3et2
differentiating both sides with respect to t, we get
dxdt=d(3et2)dt
=3detdtd2dt
=3et0
dxdt=3et
dx3=etdt
Step 2
Substituting above values, we get
etcos(3et2)dt=cos(3et2)etdt
=13cosxdx
=13(sinx)+C
Where, C is an arbitrary constant.
Hence, value of given integral is 13sinx+C.
Prioned

Prioned

Beginner2021-11-25Added 11 answers

Step 1: Use Integration by Substitution.
Let u=3et2,du=3etdt, then etdt=13du
Step 2: Using u and du above, rewrite etcos(3et2)dt.
cosu3du
Step 3: Use Constant Factor Rule: cf(x)dx=cf(x)dx.
13cosudu
Step 4: Use Trigonometric Integration: the integral of cosu is sinu.
sinu3
Step 5: Substitute u=3et2 back into the original integral.
sin(3et2)3
Step 6: Add constant.
sin(3et2)3+C

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