7.1, #6. f(x) = \sqrt{x}, so

F (x) = ∫ \sqrt{x} dx = (2/3)x^{3/2} + C.

The condition F (0) = 0 holds if C = 0. 7.1, #8. f(x) = 5 − (x^{6}/6) − x^{7}, so

F (x) = ∫ (5 − (x^{6}/6) − x^{7}) dx = 5x − (x^{7}/42) − (x^{8}/8) + C.

The condition F (0) = 0 holds if C = 0. 7.1, #9. f(x) = sin x, so

F (x) = ∫ sin x dx = − cos x + C.

The condition F (0) = 0 holds if −1 + C = 0, or equivalently, C = 1. 7.1, #12.

∫ (x^{3/2} + x^{−3/2}) dx = (2x^{5/2}/5) − 2x^{−1/2} + C

7.1, #14.

∫ \frac{y^{2} + 1}{y} dy = ∫ (y + y^{−1}) dy = \frac{y^{2}}{2} + ln |y| + C.

7.1, #16. The derivative of e^{x} is e^{x}, but e^{1} = e, so to satisfy F (1) = 0 we need to take F (x) = e^{x} − e.