What is the integral of (2x^2+3)e^{6x} ?

The integral of (2x^2+3)e^{6x} is 1/3 e^{6x}x^2-1/9 e^{6x}x+14/27 e^{6x}+C

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integral of (2x^2+3)e^{6x}

\int (2 x^{2} + 3) e^{6 x} d x

\frac{1}{3} e^{6 x} x^{2} - \frac{1}{9} e^{6 x} x + \frac{14}{27} e^{6 x} + C

Solution steps

\int (2 x^{2} + 3) e^{6 x} d x

Apply Integration By Parts

: \frac{1}{6} e^{6 x} (2 x^{2} + 3) - \int \frac{2}{3} e^{6 x} x d x

= \frac{1}{6} e^{6 x} (2 x^{2} + 3) - \int \frac{2}{3} e^{6 x} x d x

\int \frac{2}{3} e^{6 x} x d x = \frac{1}{54} (6 e^{6 x} x - e^{6 x})

= \frac{1}{6} e^{6 x} (2 x^{2} + 3) - \frac{1}{54} (6 e^{6 x} x - e^{6 x})

Simplify

\frac{1}{6} e^{6 x} (2 x^{2} + 3) - \frac{1}{54} (6 e^{6 x} x - e^{6 x}) : \frac{1}{3} e^{6 x} x^{2} - \frac{1}{9} e^{6 x} x + \frac{14}{27} e^{6 x}

= \frac{1}{3} e^{6 x} x^{2} - \frac{1}{9} e^{6 x} x + \frac{14}{27} e^{6 x}

Add a constant to the solution

= \frac{1}{3} e^{6 x} x^{2} - \frac{1}{9} e^{6 x} x + \frac{14}{27} e^{6 x} + C

\frac{d}{d x} (\frac{4 x ^{3} - 5}{x ^{2} - 3})

x \to 0 - lim (\frac{6}{x})

\frac{d}{d x} (4 x x)

\int \frac{6}{6 x + 1} d x

f (x) = sec (x), at x = - \frac{7 π}{4}

What is the integral of (2x^2+3)e^{6x} ?
The integral of (2x^2+3)e^{6x} is 1/3 e^{6x}x^2-1/9 e^{6x}x+14/27 e^{6x}+C