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