What is the integral of (3x-1)(sqrt(x)+1) ?

The integral of (3x-1)(sqrt(x)+1) is 6/5 x^{5/2}+(3x^2)/2-2/3 x^{3/2}-x+C

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integral of (3x-1)(sqrt(x)+1)

\int (3 x - 1) (x + 1) d x

\frac{6}{5} x^{\frac{5}{2}} + \frac{3 x ^{2}}{2} - \frac{2}{3} x^{\frac{3}{2}} - x + C

Solution steps

\int (3 x - 1) (x + 1) d x

Expand

(3 x - 1) (x + 1) : 3 x^{\frac{3}{2}} + 3 x - x - 1

= \int 3 x^{\frac{3}{2}} + 3 x - x - 1 d x

Apply the Sum Rule:

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

= \int 3 x^{\frac{3}{2}} d x + \int 3 x d x - \int x d x - \int 1 d x

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

\int 3 x d x = \frac{3 x ^{2}}{2}

\int x d x = \frac{2}{3} x^{\frac{3}{2}}

\int 1 d x = x

= \frac{6}{5} x^{\frac{5}{2}} + \frac{3 x ^{2}}{2} - \frac{2}{3} x^{\frac{3}{2}} - x

Add a constant to the solution

= \frac{6}{5} x^{\frac{5}{2}} + \frac{3 x ^{2}}{2} - \frac{2}{3} x^{\frac{3}{2}} - x + C

\int \frac{sin ( 3 x )}{2 + cos ( 3 x )} d x

\int a r cos (x) d x

y = x \frac{a ^{2} - x ^{2}}{x}

x = y^{2} - 12, x = 4 y

\int cos (y) e^{5 s i n (y)} d y

What is the integral of (3x-1)(sqrt(x)+1) ?
The integral of (3x-1)(sqrt(x)+1) is 6/5 x^{5/2}+(3x^2)/2-2/3 x^{3/2}-x+C