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

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

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

\int (x + 1) (x^{3} + 1) d x

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

Solution steps

\int (x + 1) (x^{3} + 1) d x

Expand

(x + 1) (x^{3} + 1) : x x^{3} + x + x^{3} + 1

= \int x x^{3} + x + x^{3} + 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 x x^{3} d x + \int x d x + \int x^{3} d x + \int 1 d x

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

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

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

\int 1 d x = x

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

Add a constant to the solution

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

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