To determine
To evaluate:
∫-23(x2-3)dx
Answer
-103
Explanation
1) Concept:
By using the fundamental theorem of calculus and the rules of integration, evaluate the given integral.
The fundamental theorem of calculus:
If f is continuous on [a, b], then ∫abfxdx=Fb-F(a).
2) Formula:
∫fx+gxdx=∫fxdx+∫gxdx
∫xndx=xn+1n+1+C
∫kdx=kx+C
3) Given:
∫-23(x2-3)dx
4) Calculation:
Consider, ∫-23(x2-3)dx, by applying rules of integral,
After separating the integrals,
∫-23(x2-3)dx=∫-23x2dx-∫-233dx
By applying the fundamental theorem of calculus and the power rule,
=x33-3x-23
=333—33--233-3-2
=273- 9--83+6
As L.C.M. of 1, 3=3. Therefore,
= 9-9--83+183
=0-103
=-103
Conclusion:
Therefore, ∫-23(x2-3)dx= -103