What is the integral of (r^2-10^r+4)/(r^3) ?

The integral of (r^2-10^r+4)/(r^3) is

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integral of (r^2-10^r+4)/(r^3)

\int \frac{r ^{2} - 1 0 ^{r} + 4}{r ^{3}} d r

ln (ln (10) ∣ r ∣) + \frac{5 ^{r} \cdot 2 ^{r - 1}}{r ^{2}} + \frac{ln ( 10 ) \cdot 5 ^{r} \cdot 2 ^{r - 1}}{r} - \frac{1}{2} ln^{2} (10) Ei (ln (1 0^{r})) - \frac{2}{r ^{2}} + C

Solution steps

\int \frac{r ^{2} - 1 0 ^{r} + 4}{r ^{3}} d r

Apply u-substitution

: \int \frac{ln ^{2} ( u ) - ln ^{2} ( 10 ) u + 4 ln ^{2} ( 10 )}{u ln ^{3} ( u )} d u

= \int \frac{ln ^{2} ( u ) - ln ^{2} ( 10 ) u + 4 ln ^{2} ( 10 )}{u ln ^{3} ( u )} d u

Expand

\frac{ln ^{2} ( u ) - ln ^{2} ( 10 ) u + 4 ln ^{2} ( 10 )}{u ln ^{3} ( u )} : \frac{1}{u ln ( u )} - \frac{ln ^{2} ( 10 )}{ln ^{3} ( u )} + \frac{4 ln ^{2} ( 10 )}{u ln ^{3} ( u )}

= \int \frac{1}{u ln ( u )} - \frac{ln ^{2} ( 10 )}{ln ^{3} ( u )} + \frac{4 ln ^{2} ( 10 )}{u ln ^{3} ( u )} d u

Apply the Sum Rule:

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

= \int \frac{1}{u ln ( u )} d u - \int \frac{ln ^{2} ( 10 )}{ln ^{3} ( u )} d u + \int \frac{4 ln ^{2} ( 10 )}{u ln ^{3} ( u )} d u

\int \frac{1}{u ln ( u )} d u = ln ∣ ln (u) ∣

\int \frac{ln ^{2} ( 10 )}{ln ^{3} ( u )} d u = ln^{2} (10) (- \frac{u}{2 ln ^{2} ( u )} + \frac{1}{2} (- \frac{u}{ln ( u )} + Ei (ln (u))))

\int \frac{4 ln ^{2} ( 10 )}{u ln ^{3} ( u )} d u = - \frac{2 ln ^{2} ( 10 )}{ln ^{2} ( u )}

= ln ∣ ln (u) ∣ - ln^{2} (10) (- \frac{u}{2 ln ^{2} ( u )} + \frac{1}{2} (- \frac{u}{ln ( u )} + Ei (ln (u)))) - \frac{2 ln ^{2} ( 10 )}{ln ^{2} ( u )}

Substitute back

u = 1 0^{r}

= ln ∣ ln (1 0^{r}) ∣ - ln^{2} (10) (- \frac{1 0 ^{r}}{2 ln ^{2} ( 1 0 ^{r} )} + \frac{1}{2} (- \frac{1 0 ^{r}}{ln ( 1 0 ^{r} )} + Ei (ln (1 0^{r})))) - \frac{2 ln ^{2} ( 10 )}{ln ^{2} ( 1 0 ^{r} )}

Simplify

ln ∣ ln (1 0^{r}) ∣ - ln^{2} (10) (- \frac{1 0 ^{r}}{2 ln ^{2} ( 1 0 ^{r} )} + \frac{1}{2} (- \frac{1 0 ^{r}}{ln ( 1 0 ^{r} )} + Ei (ln (1 0^{r})))) - \frac{2 ln ^{2} ( 10 )}{ln ^{2} ( 1 0 ^{r} )} : ln (ln (10) ∣ r ∣) + \frac{5 ^{r} \cdot 2 ^{r - 1}}{r ^{2}} + \frac{ln ( 10 ) \cdot 5 ^{r} \cdot 2 ^{r - 1}}{r} - \frac{1}{2} ln^{2} (10) Ei (ln (1 0^{r})) - \frac{2}{r ^{2}}

= ln (ln (10) ∣ r ∣) + \frac{5 ^{r} \cdot 2 ^{r - 1}}{r ^{2}} + \frac{ln ( 10 ) \cdot 5 ^{r} \cdot 2 ^{r - 1}}{r} - \frac{1}{2} ln^{2} (10) Ei (ln (1 0^{r})) - \frac{2}{r ^{2}}

Add a constant to the solution

= ln (ln (10) ∣ r ∣) + \frac{5 ^{r} \cdot 2 ^{r - 1}}{r ^{2}} + \frac{ln ( 10 ) \cdot 5 ^{r} \cdot 2 ^{r - 1}}{r} - \frac{1}{2} ln^{2} (10) Ei (ln (1 0^{r})) - \frac{2}{r ^{2}} + C

\int_{3}^{\infty} \frac{1}{x ^{2} + x} d x

\frac{d}{d x} ((3 x - 2)^{4})

x^{7} sin (x)

\int 2 x (x^{2} + 6)^{9} d x

\frac{d}{d x} (arcsin (4 - 3 x))

What is the integral of (r^2-10^r+4)/(r^3) ?
The integral of (r^2-10^r+4)/(r^3) is