Math, asked by suthar2317, 1 year ago

Integration of (2x^7+3x^6-10x^5-7x^3-12x^2+x+1)/(x^2+2)

Answers

Answered by amitnrw
1

Given : (2x^7+3x^6-10x^5-7x^3-12x^2+x+1)/(x^2+2)

To Find : Integrate

Solution:

                  2x⁵ + 3x⁴ - 14x³ - 6x² + 21x

x² +2     _|  2x⁷+3x⁶-10x⁵-7x³-12x²+x+1  |_

                  2x⁷       + 4x⁵

              ____________________

                        3x⁶-14x⁵-7x³-12x²+x+1

                        3x⁶ +6x⁴

                     _________________

                             -14x⁵ -6x⁴-7x³-12x²+x+1

                             -14x⁵       -28x³

                    ______________________

                                      -6x⁴  + 21x³  - 12x²  + x + 1

                                      -6x⁴             -12x²    

                                      ________________

                                                  21x³   + x + 1

                                                  21x³   + 42x

                                                ____________

                                                            -41x  + 1

2x⁵ + 3x⁴ - 14x³ - 6x² + 21x   - 41x/(x² +2)  + 1/(x² +2)

= 2x⁵ + 3x⁴ - 14x³ - 6x² + 21x - (41/2) 2x/(x² +2) +  1/(x² +2)

Integrate

I =  ∫ (2x⁵ + 3x⁴ - 14x³ - 6x² + 21x - (41/2) 2x/(x² +2) +  1/(x² +(√2)²)) dx

Using integration identities  

∫xⁿdx = xⁿ⁺¹ / (n + 1) + c

∫( 1/(x² +a²))dx = (1/a)tan⁻¹(x/a) + c

∫dx/x = ln | x | + c    or ∫ f'(x)dx / f(x) = ln | f(x) | + c

2x⁶/6 + 3x⁵/5 - 14x⁴/4 - 6x³/3 + 21x²/2 - (41/2)ln|x² +2|  + (1/√2)tan⁻¹(x/√2) +C

= x⁶/3+ 3x⁵/5 - 7x⁴/2 - 2x³  + 21x²/2 - (41/2)ln|x² +2|  + (1/√2)tan⁻¹(x/√2) +C

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Answered by iTzSiNgLeGiRl
0

Step-by-step explanation:

the chemical element of atomic number 7, a colourless, odourless unreactive gas that forms about 78 per cent of the earth's atmosphere. Liquid nitrogen (made by distilling liquid air) boils at 77.4 kelvins (−195.8°C) and is used as a coolant.

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