Use the methods introduced evaluate the following integrals. \int_{-2}^{1}\frac{3}{x^{2}+4x+13}dx

hadejada7x

hadejada7x

Answered question

2022-01-04

Use the methods introduced evaluate the following integrals.
213x2+4x+13dx

Answer & Explanation

Jordan Mitchell

Jordan Mitchell

Beginner2022-01-05Added 31 answers

Step 1
The given integral is 213x2+4x+13dx.
Rewrite the given integral as follows:
213x2+4x+13dx=3211x2+4x+13dx
=3211(x+2)2+9
Step 2
Apply u-substitution:
u=x+2du=dx
when x=1u=3
when x=2u=0
Now,
213x2+4x+13dx=3031u2+9dx
Apply integral substitution: u=3v
213x2+4x+13dx=30113(v2+1)dx
0213x2+4x+13dx=011(v2+1)dx
=[arctan(v)]01
=π4
Terry Ray

Terry Ray

Beginner2022-01-06Added 50 answers

213x2+4x+13dx
We calculate the integral:
31(x+2)2+9dx=arctan(13(x+2))
Answer:
arctan(13(x+2))+C
We calculate the definite integral:
213x2+4x+13dx=(arctan(13(x+2)))21
F(1)=π4
F(-2)=0
I=π4(0)=π4
karton

karton

Expert2022-01-11Added 613 answers

213x2+4x+13dx3x2+4x+13dx3x2+4x+4+9dx3(x+2)2+9dx3×1(x+2)2+9dx3×1t2+9dt3×13×arctan(t3)3×13×arctan(x+23)arctan(x+23)arctan(x+23)|21arctan(1+23)arctan(2+23)Answer:π4

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