Exercise · ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-002 · ★☆☆ Basic
Compute \(\displaystyle\int_{0}^{2} \left(3 x^{2} + 1\right)\,dx\).
Choose one option.
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- C. \(10\)
Explanation
An antiderivative is \(F(x)=x^{3} + x\). By the fundamental theorem, the integral is \(F(2)-F(0)=10\).
Review the concept: Fundamental theorem of calculus
It links derivatives and integrals: a definite integral equals an antiderivative evaluated at the endpoints, and differentiating an integral with variable upper limit returns the integrand.
\[\begin{gathered}\int_a^b f(x)\,dx=F(b)-F(a)\\ \frac{d}{dx}\int_a^x f(t)\,dt=f(x)\end{gathered}\]
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