Question

| -/1 POINTS Find the indefinite integral and check your result by differentiation. (Use C for the constant of integration.) (7-2) dr Need Help? Read It Talk to a Tutor 4. -11 POINTS Find the indefinite integral and check the result by differentiation. (Use C for the constant of integration.) 1 / (5x - sx?) dx Need Help? Read It Watch It Talk to a Tutor

Answer 1

(3)

$\int \left ( 7-x \right )dx=\int 7\, dx-\int x\, dx$

$=7\int dx-\frac{x^{2}}{2}$

$=7x-\frac{x^{2}}{2}+C$

$\frac{\mathrm{d} }{\mathrm{d} x}\left [7x-\frac{x^{2}}{2}+C \right ]=\frac{\mathrm{d} }{\mathrm{d} x}\left [7x \right ]-\frac{\mathrm{d} }{\mathrm{d} x}\left [\frac{x^{2}}{2} \right ]+\frac{\mathrm{d} }{\mathrm{d} x}\left [C \right ]$

$=7\times \frac{\mathrm{d}x }{\mathrm{d} x}-\frac{1}{2}\times \frac{\mathrm{d} }{\mathrm{d} x}\left [x^{2} \right ]+0$

$=7\times 1-\frac{1}{2}\times 2x$

$=7-x$

Hence verified the result of integral with differentiation.

-----------------------------------------------------------------------------

(4)

$\int \left ( 5x-8x^{2} \right )dx=\int 5x\, dx-\int 8x^{2} dx$

$=5\int x\, dx-8\int x^{2} dx$

$=5\times \frac{x^{2}}{2}-8\times \frac{x^{3}}{3}+C$

$= \frac{5x^{2}}{2}- \frac{8x^{3}}{3}+C$

$\frac{\mathrm{d} }{\mathrm{d} x}\left [ \frac{5x^{2}}{2}- \frac{8x^{3}}{3}+C \right ]=\frac{\mathrm{d} }{\mathrm{d} x}\left [ \frac{5x^{2}}{2} \right ]-\frac{\mathrm{d} }{\mathrm{d} x}\left [ \frac{8x^{3}}{3} \right ]+\frac{\mathrm{d} }{\mathrm{d} x}\left [ C \right ]$

$=\frac{5}{2}\times \frac{\mathrm{d} }{\mathrm{d} x}\left [ x^{2} \right ]-\frac{8}{3}\times \frac{\mathrm{d} }{\mathrm{d} x}\left [ x^{3} \right ]+0$

$=\frac{5}{2}\times 2x-\frac{8}{3}\times 3x^{2}$

$=5x-8x^{2}$

Hence verified the result of integral with differentiation.

-------------------------------------------------------------------------------