To determine
To evaluate:
∫19u-2u2udu if it exists
Answer
-76
Explanation
1) Concept:
By using fundamental theorem of calculus and rules of integration evaluate the given integral.
2) Theorem:
Fundamental theorem of calculus:
If f is continuous on [a, b], then ∫abfxdx=Fb-F(a).
3) Formula:
∫xndx=xn+1n+1+C
3) Given:
∫19u-2u2udu
4) Calculation:
Consider, ∫19u-2u2udu
Simplify,
=∫19u12u-2u2udu
=∫19(u-12-2u) du
Since the function is continuous in the given interval, the integral exists.
By applying fundamental theorem of calculus and power rule,
=u1212-2u2219
=(9)1212-(9)2-(1)1212-(1)2
Simplify,
=6-81-(2-1)
=-75-1
=-76
Conclusion:
Therefore, ∫19u-2u2udu=-76