Question

Find S (3x - 5)(x – 3)dx, with C as the constant of integration. S(3x – 5)(x – 3)dx = Enter your next step here

Answer 1

$\text{(7) }$

(3.1 – 5)(x – 3)dx = (3x2 – 9x – 5x + 15)dx

=> / (3.1 – 5) (x – 3)da = (3x2 – 14.2 +15)dx

-> |(33 – 5)(2 – 3)dx = ) (3x+)dx = |(1417)dx + |(15)da

=> ) (3x – 5) (x – 3)dx = 3 / 2?dx – 14 / xdx + 15 | dx

$=> \int (3x-5)(x-3)dx =3\left ( \frac{x^{2+1}}{(2+1)} \right )-14\left ( \frac{x^{1+1}}{(1+1)} \right )+15(x)+C$

  $\text {Where C is the constant of integration. }$

$\left ( \because \int x^n dx = \frac{x^{n+1}}{n+1} \right )$

=> | (31 – 5)(x – 3)dx = 22 – 7z? +152 +0

$\text {(8)}$

$\text {Antiderivative of }-7cosx \text { will be given by }$

$\text {Antiderivative} = \int (-7cosx)dx = -7\int( cosx)dx = -7sinx +C$

$\text {Where C is the constant of integration. }$

$=> \text {Antiderivative} = -7sinx +C$

$\text {(9) }$

$\int \left ( 5x^{\frac{2}{3} } + 6\sqrt{x}+2 \right )dx = \int ( 5x^{\frac{2}{3} })dx +\int ( 6\sqrt{x})dx+\int (2)dx$

$=> \int \left ( 5x^{\frac{2}{3} } + 6\sqrt{x}+2 \right )dx = 5\int x^{\frac{2}{3} }dx +6\int \sqrt{x}dx+2\int dx$

$=> \int \left ( 5x^{\frac{2}{3} } + 6\sqrt{x}+2 \right )dx =5\left (\frac{x^{\frac{2}{3}+1}}{\frac{2}{3}+1} \right )+6\left ( \frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1} \right )+2(x) +C$

$\text {Where C is the constant of integration. }$

$\left ( \because \int x^n dx = \frac{x^{n+1}}{n+1} \right )$

$=> \int \left ( 5x^{\frac{2}{3} } + 6\sqrt{x}+2 \right )dx =3x^{\frac{5}{3}}+4x^{\frac{3}{2}}+2x +C$

$\text{ (10)}$

$\int \frac{x^3+4}{\sqrt{x}}dx= \int \left ( \frac{x^3}{\sqrt{x}} +\frac{4}{\sqrt{x}} \right )dx$

$=> \int \frac{x^3+4}{\sqrt{x}}dx= \int\left ( x^{\frac{5}{2}} +4x^{-\frac{1}{2}} \right )dx$

$=> \int \frac{x^3+4}{\sqrt{x}}dx= \int x^{\frac{5}{2}} dx +\int (4x^{-\frac{1}{2}} )dx$

$=> \int \frac{x^3+4}{\sqrt{x}}dx= \int x^{\frac{5}{2}} dx +4\int x^{-\frac{1}{2}} dx$

$=> \int \frac{x^3+4}{\sqrt{x}}dx= \frac{x^{\frac{5}{2}+1} }{\frac{5}{2}+1} +4\left (\frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1} \right ) +C$

$\text {Where C is the constant of integration. }$

$\left ( \because \int x^n dx = \frac{x^{n+1}}{n+1} \right )$

$=> \int \frac{x^3+4}{\sqrt{x}}dx= \frac{2}{7}x^\frac{7}{2}+8x^\frac{1}{2}+C$