What is the integral of x^9(5x^{10}+1)^{30} ?

The integral of x^9(5x^{10}+1)^{30} is 1/1550 (5x^{10}+1)^{31}+C

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integral of x^9(5x^{10}+1)^{30}

\int x^{9} (5 x^{10} + 1)^{30} d x

\frac{1}{1550} (5 x^{10} + 1)^{31} + C

Solution steps

\int x^{9} (5 x^{10} + 1)^{30} d x

Apply u-substitution

: \int \frac{u ^{30}}{50} d u

= \int \frac{u ^{30}}{50} d u

Take the constant out:

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

= \frac{1}{50} \cdot \int u^{30} d u

Apply the Power Rule

: \frac{u ^{31}}{31}

= \frac{1}{50} \cdot \frac{u ^{31}}{31}

Substitute back

u = 5 x^{10} + 1

= \frac{1}{50} \cdot \frac{( 5 x ^{10} + 1 ) ^{31}}{31}

Simplify

\frac{1}{50} \cdot \frac{( 5 x ^{10} + 1 ) ^{31}}{31} : \frac{1}{1550} (5 x^{10} + 1)^{31}

= \frac{1}{1550} (5 x^{10} + 1)^{31}

Add a constant to the solution

= \frac{1}{1550} (5 x^{10} + 1)^{31} + C

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What is the integral of x^9(5x^{10}+1)^{30} ?
The integral of x^9(5x^{10}+1)^{30} is 1/1550 (5x^{10}+1)^{31}+C