What is the integral of (x^2+2x)e^{x^3+3x^2-1} ?

The integral of (x^2+2x)e^{x^3+3x^2-1} is 1/3 e^{x^3+3x^2-1}+C

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integral of (x^2+2x)e^{x^3+3x^2-1}

\int (x^{2} + 2 x) e^{x^{3} + 3 x^{2} - 1} d x

\frac{1}{3} e^{x^{3} + 3 x^{2} - 1} + C

Solution steps

\int (x^{2} + 2 x) e^{x^{3} + 3 x^{2} - 1} d x

Factor out common term

x : x (x + 2)

= \int e^{x^{3} + 3 x^{2} - 1} x (x + 2) d x

Apply u-substitution

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

= \int \frac{e ^{u}}{3} d u

Take the constant out:

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

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

Use the common integral:

\int e^{u} d u = e^{u}

= \frac{1}{3} e^{u}

Substitute back

u = x^{3} + 3 x^{2} - 1

= \frac{1}{3} e^{x^{3} + 3 x^{2} - 1}

Add a constant to the solution

= \frac{1}{3} e^{x^{3} + 3 x^{2} - 1} + C

f (x) = (x - 1)^{2} + 1, at x = - 1

f (x) = - 10 x + 5, (1, 2)

\int x^{2} 4 x^{2} + 1 d x

x \to 3 lim (\frac{x ^{2}}{x})

\frac{d}{d x} (\frac{sin ( 3 x + 1 )}{7 x ^{2} + 1})

What is the integral of (x^2+2x)e^{x^3+3x^2-1} ?
The integral of (x^2+2x)e^{x^3+3x^2-1} is 1/3 e^{x^3+3x^2-1}+C