What is the integral of (sqrt(x)+2)(x^2+sqrt(x)-1) ?

The integral of (sqrt(x)+2)(x^2+sqrt(x)-1) is 2/7 x^{7/2}+(2x^3)/3+(x^2)/2+2/3 x^{3/2}-2x+C

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integral of (sqrt(x)+2)(x^2+sqrt(x)-1)

\int (x + 2) (x^{2} + x - 1) d x

\frac{2}{7} x^{\frac{7}{2}} + \frac{2 x ^{3}}{3} + \frac{x ^{2}}{2} + \frac{2}{3} x^{\frac{3}{2}} - 2 x + C

Solution steps

\int (x + 2) (x^{2} + x - 1) d x

Expand

(x + 2) (x^{2} + x - 1) : x^{\frac{5}{2}} + 2 x^{2} + x + x - 2

= \int x^{\frac{5}{2}} + 2 x^{2} + x + x - 2 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 x^{\frac{5}{2}} d x + \int 2 x^{2} d x + \int x d x + \int x d x - \int 2 d x

\int x^{\frac{5}{2}} d x = \frac{2}{7} x^{\frac{7}{2}}

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

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

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

\int 2 d x = 2 x

= \frac{2}{7} x^{\frac{7}{2}} + \frac{2 x ^{3}}{3} + \frac{x ^{2}}{2} + \frac{2}{3} x^{\frac{3}{2}} - 2 x

Add a constant to the solution

= \frac{2}{7} x^{\frac{7}{2}} + \frac{2 x ^{3}}{3} + \frac{x ^{2}}{2} + \frac{2}{3} x^{\frac{3}{2}} - 2 x + C

t \to 0 lim (\frac{5}{t} - \frac{5}{t ^{2} + t})

n \to \infty lim (\frac{n ^{100}}{5 0 ^{n}})

\frac{d}{d x} (sec (arctan (x)))

\frac{x ^{5}}{3 - x ^{4}}

ln (ln (\frac{x}{2 - x ^{2}}))

What is the integral of (sqrt(x)+2)(x^2+sqrt(x)-1) ?
The integral of (sqrt(x)+2)(x^2+sqrt(x)-1) is 2/7 x^{7/2}+(2x^3)/3+(x^2)/2+2/3 x^{3/2}-2x+C