What is the integral of 42x(x+7)^5 ?

The integral of 42x(x+7)^5 is 42((x^7)/7+(35x^6)/6+98x^5+(1715x^4)/2+(12005x^3)/3+(16807x^2)/2)+C

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integral of 42x(x+7)^5

\int 42 x (x + 7)^{5} d x

42 (\frac{x ^{7}}{7} + \frac{35 x ^{6}}{6} + 98 x^{5} + \frac{1715 x ^{4}}{2} + \frac{12005 x ^{3}}{3} + \frac{16807 x ^{2}}{2}) + C

Solution steps

\int 42 x (x + 7)^{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 + 7)^{5} d x

Expand

x (x + 7)^{5} : x^{6} + 35 x^{5} + 490 x^{4} + 3430 x^{3} + 12005 x^{2} + 16807 x

= 42 \cdot \int x^{6} + 35 x^{5} + 490 x^{4} + 3430 x^{3} + 12005 x^{2} + 16807 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 35 x^{5} d x + \int 490 x^{4} d x + \int 3430 x^{3} d x + \int 12005 x^{2} d x + \int 16807 x d x)

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

\int 35 x^{5} d x = \frac{35 x ^{6}}{6}

\int 490 x^{4} d x = 98 x^{5}

\int 3430 x^{3} d x = \frac{1715 x ^{4}}{2}

\int 12005 x^{2} d x = \frac{12005 x ^{3}}{3}

\int 16807 x d x = \frac{16807 x ^{2}}{2}

= 42 (\frac{x ^{7}}{7} + \frac{35 x ^{6}}{6} + 98 x^{5} + \frac{1715 x ^{4}}{2} + \frac{12005 x ^{3}}{3} + \frac{16807 x ^{2}}{2})

Add a constant to the solution

= 42 (\frac{x ^{7}}{7} + \frac{35 x ^{6}}{6} + 98 x^{5} + \frac{1715 x ^{4}}{2} + \frac{12005 x ^{3}}{3} + \frac{16807 x ^{2}}{2}) + C

\frac{d}{d x} (26 (2 e^{x^{2}} x^{2} + e^{x^{2}}))

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\int e^{t} (sin (t) + cos (t)) d t

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What is the integral of 42x(x+7)^5 ?
The integral of 42x(x+7)^5 is 42((x^7)/7+(35x^6)/6+98x^5+(1715x^4)/2+(12005x^3)/3+(16807x^2)/2)+C