What is the integral of (-x^{pi/3}-3x^{sqrt(3)}+5) ?

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

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integral of (-x^{pi/3}-3x^{sqrt(3)}+5)

\int (- x^{\frac{π}{3}} - 3 x^{3} + 5) d x

- \frac{3}{π + 3} x^{\frac{π + 3}{3}} - \frac{3}{3 + 1} x^{3 + 1} + 5 x + C

Solution steps

\int - x^{\frac{π}{3}} - 3 x^{3} + 5 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{π}{3}} d x - \int 3 x^{3} d x + \int 5 d x

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

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

\int 5 d x = 5 x

= - \frac{3}{π + 3} x^{\frac{π + 3}{3}} - \frac{3}{3 + 1} x^{3 + 1} + 5 x

Add a constant to the solution

= - \frac{3}{π + 3} x^{\frac{π + 3}{3}} - \frac{3}{3 + 1} x^{3 + 1} + 5 x + C

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