Integration

1. ∫2x^2 - 4x^-3 dx
= (2/3)x^3 - 4/(-2) x^(-2) + C, where C is constant
= (2/3)x^3 + 2x^(-2) + C
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2. ∫cos(ax + b) dx ,where a and b are any real numbers
= (1/a)∫cos(ax + b) d(ax + b)
= (1/a) sin(ax + b) + C, where C is constant
when a = 0, ∫cos(ax + b) dx = ∫cos(b) dx = x cos(b) + C
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3. ∫( 4 + 1/x )^2 dx
= ∫( 16 + 8/x + 1/x^2 ) dx
= 16x + 8lnx - 1/x + C
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4. ∫2/x^2 - 1 dx by partial fraction method
( Hint : 2/x^2 - 1 = A/x-1 + B/x+1 )
∫2/(x^2 - 1) dx
= ∫1/(x-1) - 1/(x+1) dx
= ln(x-1) - ln(x+1) + C