What is the integral of (8x+10)(7-2x^2-5x)^3 ?

The integral of (8x+10)(7-2x^2-5x)^3 is -1/2097152 (-64x^2-160x+224)^4+C

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integral of (8x+10)(7-2x^2-5x)^3

\int (8 x + 10) (7 - 2 x^{2} - 5 x)^{3} d x

- \frac{1}{2097152} (- 64 x^{2} - 160 x + 224)^{4} + C

Solution steps

\int (8 x + 10) (7 - 2 x^{2} - 5 x)^{3} d x

Apply u-substitution

: \int \frac{u ( - u ^{2} + 324 ) ^{3}}{262144} d u

= \int \frac{u ( - u ^{2} + 324 ) ^{3}}{262144} d u

Take the constant out:

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

= \frac{1}{262144} \cdot \int u (- u^{2} + 324)^{3} d u

Apply u-substitution

: \int - \frac{v ^{3}}{2} d v

= \frac{1}{262144} \cdot \int - \frac{v ^{3}}{2} d v

Take the constant out:

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

= \frac{1}{262144} (- \frac{1}{2} \cdot \int v^{3} d v)

Apply the Power Rule

: \frac{v ^{4}}{4}

= \frac{1}{262144} (- \frac{1}{2} \cdot \frac{v ^{4}}{4})

Substitute back

= \frac{1}{262144} - \frac{1}{2} \cdot \frac{( - ( 8 x + 10 ) ^{2} + 324 ) ^{4}}{4}

Simplify

\frac{1}{262144} - \frac{1}{2} \cdot \frac{( - ( 8 x + 10 ) ^{2} + 324 ) ^{4}}{4} : - \frac{1}{2097152} (- 64 x^{2} - 160 x + 224)^{4}

= - \frac{1}{2097152} (- 64 x^{2} - 160 x + 224)^{4}

Add a constant to the solution

= - \frac{1}{2097152} (- 64 x^{2} - 160 x + 224)^{4} + C

\int - 0.08 x + 64 d x

\int x^{3} 8 + x^{4} d x

\int \frac{x}{1 - 6 x ^{4}} d x

\int \frac{1}{1 - p} d p

\int x^{3} \frac{1}{1 + x ^{2}} d x

What is the integral of (8x+10)(7-2x^2-5x)^3 ?
The integral of (8x+10)(7-2x^2-5x)^3 is -1/2097152 (-64x^2-160x+224)^4+C