What is the integral of 42x(x+12)^5 ?

The integral of 42x(x+12)^5 is 42((x^7)/7+10x^6+288x^5+4320x^4+34560x^3+124416x^2)+C

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integral of 42x(x+12)^5

\int 42 x (x + 12)^{5} d x

42 (\frac{x ^{7}}{7} + 10 x^{6} + 288 x^{5} + 4320 x^{4} + 34560 x^{3} + 124416 x^{2}) + C

Solution steps

\int 42 x (x + 12)^{5} d x

Take the constant out:

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

= 42 \cdot \int x (x + 12)^{5} d x

Expand

x (x + 12)^{5} : x^{6} + 60 x^{5} + 1440 x^{4} + 17280 x^{3} + 103680 x^{2} + 248832 x

= 42 \cdot \int x^{6} + 60 x^{5} + 1440 x^{4} + 17280 x^{3} + 103680 x^{2} + 248832 x 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

= 42 (\int x^{6} d x + \int 60 x^{5} d x + \int 1440 x^{4} d x + \int 17280 x^{3} d x + \int 103680 x^{2} d x + \int 248832 x d x)

\int x^{6} d x = \frac{x ^{7}}{7}

\int 60 x^{5} d x = 10 x^{6}

\int 1440 x^{4} d x = 288 x^{5}

\int 17280 x^{3} d x = 4320 x^{4}

\int 103680 x^{2} d x = 34560 x^{3}

\int 248832 x d x = 124416 x^{2}

= 42 (\frac{x ^{7}}{7} + 10 x^{6} + 288 x^{5} + 4320 x^{4} + 34560 x^{3} + 124416 x^{2})

Add a constant to the solution

= 42 (\frac{x ^{7}}{7} + 10 x^{6} + 288 x^{5} + 4320 x^{4} + 34560 x^{3} + 124416 x^{2}) + C

f (x) = \frac{5 x}{x - 3}, at x = 4

\frac{\partial}{\partial x} (x - y - 1)

f (x) = 5 x - 4

y = x + 3 x

\frac{d}{d x} (e^{x y} ln (z))

What is the integral of 42x(x+12)^5 ?
The integral of 42x(x+12)^5 is 42((x^7)/7+10x^6+288x^5+4320x^4+34560x^3+124416x^2)+C