What is the integral of r^2((r^3)/(24)-6)^7 ?

The integral of r^2((r^3)/(24)-6)^7 is ((r^3)/(24)-6)^8+C

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integral of r^2((r^3)/(24)-6)^7

\int r^{2} (\frac{r ^{3}}{24} - 6)^{7} d r

(\frac{r ^{3}}{24} - 6)^{8} + C

Solution steps

\int r^{2} (\frac{r ^{3}}{24} - 6)^{7} d r

Apply u-substitution

: \int 8 u^{7} d u

= \int 8 u^{7} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 8 \cdot \int u^{7} d u

Apply the Power Rule

: \frac{u ^{8}}{8}

= 8 \cdot \frac{u ^{8}}{8}

Substitute back

u = \frac{r ^{3}}{24} - 6

= 8 \cdot \frac{( \frac{r ^{3}}{24} - 6 ) ^{8}}{8}

Simplify

8 \cdot \frac{( \frac{r ^{3}}{24} - 6 ) ^{8}}{8} : (\frac{r ^{3}}{24} - 6)^{8}

= (\frac{r ^{3}}{24} - 6)^{8}

Add a constant to the solution

= (\frac{r ^{3}}{24} - 6)^{8} + C

sin (t + 2)

f (x) = \frac{6}{x - 1}, (3, 3)

y = 6 x^{2} - 7 x + 4, (1, 3)

sin (3 x) cos (2 x)

\int sin^{4} (2 x) cos^{2} (2 x) d x

What is the integral of r^2((r^3)/(24)-6)^7 ?
The integral of r^2((r^3)/(24)-6)^7 is ((r^3)/(24)-6)^8+C