What is the integral of x(e^{(3x-2)}+5) ?

The integral of x(e^{(3x-2)}+5) is 1/3 e^{3x-2}x-1/9 e^{3x-2}+(5x^2)/2+C

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integral of x(e^{(3x-2)}+5)

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

\frac{1}{3} e^{3 x - 2} x - \frac{1}{9} e^{3 x - 2} + \frac{5 x ^{2}}{2} + C

Solution steps

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

Expand

x (e^{3 x - 2} + 5) : e^{3 x - 2} x + 5 x

= \int e^{3 x - 2} x + 5 x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int e^{3 x - 2} x d x + \int 5 x d x

\int e^{3 x - 2} x d x = \frac{1}{3} e^{3 x - 2} x - \frac{1}{9} e^{3 x - 2}

\int 5 x d x = \frac{5 x ^{2}}{2}

= \frac{1}{3} e^{3 x - 2} x - \frac{1}{9} e^{3 x - 2} + \frac{5 x ^{2}}{2}

Add a constant to the solution

= \frac{1}{3} e^{3 x - 2} x - \frac{1}{9} e^{3 x - 2} + \frac{5 x ^{2}}{2} + C

tan (θ)

x \to 9 + lim (\frac{∣ x - 9 ∣}{x - 9})

x \to 0 lim (cot (\frac{π}{2} - x))

\int 4 sin (x) cos^{2} (x) d x

\int_{2}^{3} \frac{28}{3 - x} d x

What is the integral of x(e^{(3x-2)}+5) ?
The integral of x(e^{(3x-2)}+5) is 1/3 e^{3x-2}x-1/9 e^{3x-2}+(5x^2)/2+C