What is the integral of (x+3)^{50}x ?

The integral of (x+3)^{50}x is ((x+3)^{52})/(52)-((x+3)^{51})/(17)+C

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integral of (x+3)^{50}x

\int (x + 3)^{50} x d x

\frac{( x + 3 ) ^{52}}{52} - \frac{( x + 3 ) ^{51}}{17} + C

Solution steps

\int (x + 3)^{50} x d x

Apply u-substitution

= \int (u - 3) u^{50} d u

Expand

(u - 3) u^{50} : u^{51} - 3 u^{50}

= \int u^{51} - 3 u^{50} 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 u^{51} d u - \int 3 u^{50} d u

\int u^{51} d u = \frac{u ^{52}}{52}

\int 3 u^{50} d u = \frac{u ^{51}}{17}

= \frac{u ^{52}}{52} - \frac{u ^{51}}{17}

Substitute back

u = x + 3

= \frac{( x + 3 ) ^{52}}{52} - \frac{( x + 3 ) ^{51}}{17}

Add a constant to the solution

= \frac{( x + 3 ) ^{52}}{52} - \frac{( x + 3 ) ^{51}}{17} + C

What is the integral of (x+3)^{50}x ?
The integral of (x+3)^{50}x is ((x+3)^{52})/(52)-((x+3)^{51})/(17)+C