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