What is the integral of (3-5x)^2 ?

The integral of (3-5x)^2 is -1/15 (3-5x)^3+C

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integral of (3-5x)^2

\int (3 - 5 x)^{2} d x

- \frac{1}{15} (3 - 5 x)^{3} + C

Solution steps

\int (3 - 5 x)^{2} d x

Apply u-substitution

: \int - \frac{u ^{2}}{5} d u

= \int - \frac{u ^{2}}{5} d u

Take the constant out:

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

= - \frac{1}{5} \cdot \int u^{2} d u

Apply the Power Rule

: \frac{u ^{3}}{3}

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

Substitute back

u = 3 - 5 x

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

Simplify

- \frac{1}{5} \cdot \frac{( 3 - 5 x ) ^{3}}{3} : - \frac{1}{15} (3 - 5 x)^{3}

= - \frac{1}{15} (3 - 5 x)^{3}

Add a constant to the solution

= - \frac{1}{15} (3 - 5 x)^{3} + C

\frac{d y}{d x} = 2 y cos (x)

x \to 4 lim (- x^{2} - 9 x - 8)

\frac{d}{d x} (\frac{5 x}{2 x ^{- 2}})

x \to \infty lim (3 + x^{2})

2 y = 3 x, y = 4, 2 y + 3 x = 6