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

The integral of 3x^4(-x^5+5)^4 is -3/25 (5-x^5)^5+C

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integral of 3x^4(-x^5+5)^4

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

- \frac{3}{25} (5 - x^{5})^{5} + C

Solution steps

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

Apply u-substitution

= 3 \cdot \int - \frac{u ^{4}}{5} d u

Take the constant out:

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

= 3 (- \frac{1}{5} \cdot \int u^{4} d u)

Apply the Power Rule

= 3 (- \frac{1}{5} \cdot \frac{u ^{5}}{5})

Substitute back

u = 5 - x^{5}

= 3 (- \frac{1}{5} \cdot \frac{( 5 - x ^{5} ) ^{5}}{5})

Simplify

3 (- \frac{1}{5} \cdot \frac{( 5 - x ^{5} ) ^{5}}{5}) : - \frac{3}{25} (5 - x^{5})^{5}

= - \frac{3}{25} (5 - x^{5})^{5}

Add a constant to the solution

= - \frac{3}{25} (5 - x^{5})^{5} + C

l a pl a ce t r an s f or m t^{(\frac{3}{2})}

p a r t ia l f r a c t i o n \frac{x ^{2}}{x - 5}

\int \frac{3 - e ^{2 x}}{e ^{x}} d x

\frac{d y}{d x} = x \cdot y^{2}

\int \frac{32 x ^{2}}{x ^{4} - 34 x ^{2} + 225} d x

What is the integral of 3x^4(-x^5+5)^4 ?
The integral of 3x^4(-x^5+5)^4 is -3/25 (5-x^5)^5+C