Use a table of integrals to find the indefinite integral. \int

Cheexorgeny

Cheexorgeny

Answered question

2021-12-19

Use a table of integrals to find the indefinite integral.
cosxsin2x+1dx

Answer & Explanation

Wendy Boykin

Wendy Boykin

Beginner2021-12-20Added 35 answers

Step 1
Integration is summation of discrete data. The integral is calculated for the functions to find their area, displacement, volume, that occurs due to combination of small data.
Integration is of two types definite integral and indefinite integral. Indefinite integral are defined where upper and lower limits are not given, whereas in definite integral both upper and lower limit are there.
Step 2
The given integrand is cosxsin2x+1dx, let u=sinx differentiate it with respect to x from both sides.
u=sinx
dudx=dsinxdx
dudx=cosx
du=cosxdx...(1)
Step 3
Now, substitute value of u=sinx and du from equation (1) in cosxsin2x+1dx
cosxsin2x+1dx=duu2+1
=ln|u2+1+u|+C (using duu2+1=ln|u2+1+u|)
=ln|sin2(x)+1+sin(x)|+C (substitute u=sin(x))
Therefore, value of cosxsin2x+1dx is equal to ln|sin2(x)+1+sin(x)|+C
Edward Patten

Edward Patten

Beginner2021-12-21Added 38 answers

cos(x)sin2(x)+1dx
Substitution u=sin(x)dudx=cos(x)dx=1cos(x)du:
=1u2+1du
Lets
nick1337

nick1337

Expert2021-12-28Added 777 answers

cos(x)sin(x)2+1dx
Transform the expression
1t2+1dt
Evaluate the integral
ln(|t+t2+12|)
Substitute back
ln(|sin(x)+sin(x)2+12|)
Evaluate the power
ln(|sin(x)+sin(x)2+12|)
Add CR
Solution
ln(|sin(x)+sin(x)2+12|)+C

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