What is the integral of sqrt(1+5x^3)(15x^2) ?

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

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integral of sqrt(1+5x^3)(15x^2)

\int 1 + 5 x^{3} (15 x^{2}) d x

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

Solution steps

\int 1 + 5 x^{3} \cdot 15 x^{2} d x

Take the constant out:

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

= 15 \cdot \int 1 + 5 x^{3} x^{2} d x

Apply u-substitution

: \int \frac{2 u ^{2}}{15} d u

= 15 \cdot \int \frac{2 u ^{2}}{15} d u

Take the constant out:

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

= 15 \cdot \frac{2}{15} \cdot \int u^{2} d u

Apply the Power Rule

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

= 15 \cdot \frac{2}{15} \cdot \frac{u ^{3}}{3}

Substitute back

u = 1 + 5 x^{3}

= 15 \cdot \frac{2}{15} \cdot \frac{( 1 + 5 x ^{3} ) ^{3}}{3}

Simplify

15 \cdot \frac{2}{15} \cdot \frac{( 1 + 5 x ^{3} ) ^{3}}{3} : \frac{2}{3} (1 + 5 x^{3})^{\frac{3}{2}}

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

Add a constant to the solution

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

\int \frac{c _{1}}{x cos ^{2} ( ln ( x ) )} d x

x \to 1 lim (- x + 1)

\frac{d}{d x} (\frac{a e ^{\frac{x}{2}}}{2} - b e^{- x})

\frac{d}{d x} (\frac{1}{- 25 x ^{2} + 2 y ^{2} - 50})

x \to 4 lim (x + x)

What is the integral of sqrt(1+5x^3)(15x^2) ?
The integral of sqrt(1+5x^3)(15x^2) is 2/3 (1+5x^3)^{3/2}+C