What is the integral of x(3x+10)^{49} ?

The integral of x(3x+10)^{49} is 1/9 (((3x+10)^{51}}{51}-\frac{(3x+10)^{50})/5)+C

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integral of x(3x+10)^{49}

\int x (3 x + 10)^{49} d x

\frac{1}{9} (\frac{( 3 x + 10 ) ^{51}}{51} - \frac{( 3 x + 10 ) ^{50}}{5}) + C

Solution steps

\int x (3 x + 10)^{49} d x

Apply u-substitution

: \int \frac{u ^{49} ( u - 10 )}{9} d u

= \int \frac{u ^{49} ( u - 10 )}{9} d u

Take the constant out:

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

= \frac{1}{9} \cdot \int u^{49} (u - 10) d u

Expand

u^{49} (u - 10) : u^{50} - 10 u^{49}

= \frac{1}{9} \cdot \int u^{50} - 10 u^{49} 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

= \frac{1}{9} (\int u^{50} d u - \int 10 u^{49} d u)

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

\int 10 u^{49} d u = \frac{u ^{50}}{5}

= \frac{1}{9} (\frac{u ^{51}}{51} - \frac{u ^{50}}{5})

Substitute back

u = 3 x + 10

= \frac{1}{9} (\frac{( 3 x + 10 ) ^{51}}{51} - \frac{( 3 x + 10 ) ^{50}}{5})

Add a constant to the solution

= \frac{1}{9} (\frac{( 3 x + 10 ) ^{51}}{51} - \frac{( 3 x + 10 ) ^{50}}{5}) + C

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\int - 13 cos (x) d x

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What is the integral of x(3x+10)^{49} ?
The integral of x(3x+10)^{49} is 1/9 (((3x+10)^{51}}{51}-\frac{(3x+10)^{50})/5)+C