What is the integral of x(sqrt(x)-7x^4+1) ?

The integral of x(sqrt(x)-7x^4+1) is 2/5 x^{5/2}-(7x^6)/6+(x^2)/2+C

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integral of x(sqrt(x)-7x^4+1)

\int x (x - 7 x^{4} + 1) d x

\frac{2}{5} x^{\frac{5}{2}} - \frac{7 x ^{6}}{6} + \frac{x ^{2}}{2} + C

Solution steps

\int x (x - 7 x^{4} + 1) d x

Expand

x (x - 7 x^{4} + 1) : x^{\frac{3}{2}} - 7 x^{5} + x

= \int x^{\frac{3}{2}} - 7 x^{5} + x 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{3}{2}} d x - \int 7 x^{5} d x + \int x d x

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

\int 7 x^{5} d x = \frac{7 x ^{6}}{6}

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

= \frac{2}{5} x^{\frac{5}{2}} - \frac{7 x ^{6}}{6} + \frac{x ^{2}}{2}

Add a constant to the solution

= \frac{2}{5} x^{\frac{5}{2}} - \frac{7 x ^{6}}{6} + \frac{x ^{2}}{2} + C

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What is the integral of x(sqrt(x)-7x^4+1) ?
The integral of x(sqrt(x)-7x^4+1) is 2/5 x^{5/2}-(7x^6)/6+(x^2)/2+C