What is the integral of 3x^4(2x^5+1)^3 ?

The integral of 3x^4(2x^5+1)^3 is 3/40 (2x^5+1)^4+C

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integral of 3x^4(2x^5+1)^3

\int 3 x^{4} (2 x^{5} + 1)^{3} d x

\frac{3}{40} (2 x^{5} + 1)^{4} + C

Solution steps

\int 3 x^{4} (2 x^{5} + 1)^{3} d x

Take the constant out:

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

= 3 \cdot \int x^{4} (2 x^{5} + 1)^{3} d x

Apply u-substitution

: \int \frac{u ^{3}}{10} d u

= 3 \cdot \int \frac{u ^{3}}{10} d u

Take the constant out:

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

= 3 \cdot \frac{1}{10} \cdot \int u^{3} d u

Apply the Power Rule

: \frac{u ^{4}}{4}

= 3 \cdot \frac{1}{10} \cdot \frac{u ^{4}}{4}

Substitute back

u = 2 x^{5} + 1

= 3 \cdot \frac{1}{10} \cdot \frac{( 2 x ^{5} + 1 ) ^{4}}{4}

Simplify

3 \cdot \frac{1}{10} \cdot \frac{( 2 x ^{5} + 1 ) ^{4}}{4} : \frac{3}{40} (2 x^{5} + 1)^{4}

= \frac{3}{40} (2 x^{5} + 1)^{4}

Add a constant to the solution

= \frac{3}{40} (2 x^{5} + 1)^{4} + C

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What is the integral of 3x^4(2x^5+1)^3 ?
The integral of 3x^4(2x^5+1)^3 is 3/40 (2x^5+1)^4+C