What is the integral of sqrt(t)(t^4+t-5) ?

The integral of sqrt(t)(t^4+t-5) is 2/11 t^{11/2}+2/5 t^{5/2}-10/3 t^{3/2}+C

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integral of sqrt(t)(t^4+t-5)

\int t (t^{4} + t - 5) d t

\frac{2}{11} t^{\frac{11}{2}} + \frac{2}{5} t^{\frac{5}{2}} - \frac{10}{3} t^{\frac{3}{2}} + C

Solution steps

\int t (t^{4} + t - 5) d t

Expand

t (t^{4} + t - 5) : t^{\frac{9}{2}} + t^{\frac{3}{2}} - 5 t

= \int t^{\frac{9}{2}} + t^{\frac{3}{2}} - 5 t d t

Apply the Sum Rule:

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

= \int t^{\frac{9}{2}} d t + \int t^{\frac{3}{2}} d t - \int 5 t d t

\int t^{\frac{9}{2}} d t = \frac{2}{11} t^{\frac{11}{2}}

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

\int 5 t d t = \frac{10}{3} t^{\frac{3}{2}}

= \frac{2}{11} t^{\frac{11}{2}} + \frac{2}{5} t^{\frac{5}{2}} - \frac{10}{3} t^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{11} t^{\frac{11}{2}} + \frac{2}{5} t^{\frac{5}{2}} - \frac{10}{3} t^{\frac{3}{2}} + C

\frac{\partial}{\partial x} (2 y - \frac{3}{y ^{2}} + x)

(x^{6} - 5) x

\frac{d}{d x} (\frac{cos ( x )}{x + 1})

\frac{d}{d x} (\frac{1}{x + k})

(x - 3) (x + 2) (x + 3)

What is the integral of sqrt(t)(t^4+t-5) ?
The integral of sqrt(t)(t^4+t-5) is 2/11 t^{11/2}+2/5 t^{5/2}-10/3 t^{3/2}+C