What is the integral of (1+3z)(2sqrt(z)-5) ?

The integral of (1+3z)(2sqrt(z)-5) is 4/3 z^{3/2}-5z+12/5 z^{5/2}-(15z^2)/2+C

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integral of (1+3z)(2sqrt(z)-5)

\int (1 + 3 z) (2 z - 5) d z

\frac{4}{3} z^{\frac{3}{2}} - 5 z + \frac{12}{5} z^{\frac{5}{2}} - \frac{15 z ^{2}}{2} + C

Solution steps

\int (1 + 3 z) (2 z - 5) d z

Expand

(1 + 3 z) (2 z - 5) : 2 z - 5 + 6 z^{\frac{3}{2}} - 15 z

= \int 2 z - 5 + 6 z^{\frac{3}{2}} - 15 z d z

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 2 z d z - \int 5 d z + \int 6 z^{\frac{3}{2}} d z - \int 15 z d z

\int 2 z d z = \frac{4}{3} z^{\frac{3}{2}}

\int 5 d z = 5 z

\int 6 z^{\frac{3}{2}} d z = \frac{12}{5} z^{\frac{5}{2}}

\int 15 z d z = \frac{15 z ^{2}}{2}

= \frac{4}{3} z^{\frac{3}{2}} - 5 z + \frac{12}{5} z^{\frac{5}{2}} - \frac{15 z ^{2}}{2}

Add a constant to the solution

= \frac{4}{3} z^{\frac{3}{2}} - 5 z + \frac{12}{5} z^{\frac{5}{2}} - \frac{15 z ^{2}}{2} + C

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What is the integral of (1+3z)(2sqrt(z)-5) ?
The integral of (1+3z)(2sqrt(z)-5) is 4/3 z^{3/2}-5z+12/5 z^{5/2}-(15z^2)/2+C