What is the integral of sqrt(t)(t^6+t-5) ?

The integral of sqrt(t)(t^6+t-5) is 2/15 t^{15/2}+2/5 t^{5/2}-10/3 t^{3/2}+C

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integral of sqrt(t)(t^6+t-5)

\int t (t^{6} + t - 5) d t

\frac{2}{15} t^{\frac{15}{2}} + \frac{2}{5} t^{\frac{5}{2}} - \frac{10}{3} t^{\frac{3}{2}} + C

Solution steps

\int t (t^{6} + t - 5) d t

Expand

t (t^{6} + t - 5) : t^{\frac{13}{2}} + t^{\frac{3}{2}} - 5 t

= \int t^{\frac{13}{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{13}{2}} d t + \int t^{\frac{3}{2}} d t - \int 5 t d t

\int t^{\frac{13}{2}} d t = \frac{2}{15} t^{\frac{15}{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}{15} t^{\frac{15}{2}} + \frac{2}{5} t^{\frac{5}{2}} - \frac{10}{3} t^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{15} t^{\frac{15}{2}} + \frac{2}{5} t^{\frac{5}{2}} - \frac{10}{3} t^{\frac{3}{2}} + C

x \to \infty lim (3 e^{- 3 x})

d er i v a t i v e f (x) = (x^{3} - 5 x) (5 x^{2} + 3 x)

\frac{\partial}{\partial x} ([A x^{2} + B y^{3}]^{\frac{1}{4}})

\frac{d}{d x} (\frac{6}{x} - 2)

\int_{0}^{5} π (2 7 y)^{2} d y

What is the integral of sqrt(t)(t^6+t-5) ?
The integral of sqrt(t)(t^6+t-5) is 2/15 t^{15/2}+2/5 t^{5/2}-10/3 t^{3/2}+C