Antiderivative and the indefinite integral
Answers are for parents and teachers.
1
Prove that F(x) = 3x² − sin x + 5 is an antiderivative of f(x) = 6x − cos x.
F′ = 6x − cos x = f(x).
2
Write all antiderivatives of f(x) = 6x² − 4x + 1.
F(x) = 2x³ − 2x² + x + C.
3
Find the antiderivative of f(x) = 3x² − 2x whose graph passes through (2, 5).
F = x³ − x² + C; 8 − 4 + C = 5 ⇒ C = 1; F(x) = x³ − x² + 1.
4
A point has velocity v(t) = 3t² + 4 (m/s) and s(0) = 2 m. Find s(t) and compute s(2).
s = t³ + 4t + C, C = 2; s(2) = 8 + 8 + 2 = 18 m.
5
Find an antiderivative of f(x) = 1/x² (x > 0). Hint: 1/x² = x⁻².
F(x) = −1/x + C, since (−1/x)′ = 1/x².