What is the integral of (2x+3)e^{0.5x} ?

The integral of (2x+3)e^{0.5x} is 4e^{0.5x}x-2e^{0.5x}+C

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integral of (2x+3)e^{0.5x}

\int (2 x + 3) e^{0.5 x} d x

4 e^{0.5 x} x - 2 e^{0.5 x} + C

Solution steps

\int (2 x + 3) e^{0.5 x} d x

Expand

(2 x + 3) e^{0.5 x} : 2 e^{0.5 x} x + 3 e^{0.5 x}

= \int 2 e^{0.5 x} x + 3 e^{0.5 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 2 e^{0.5 x} x d x + \int 3 e^{0.5 x} d x

\int 2 e^{0.5 x} x d x = 2 (2 e^{0.5 x} x - 4 e^{0.5 x})

\int 3 e^{0.5 x} d x = 6 e^{0.5 x}

= 2 (2 e^{0.5 x} x - 4 e^{0.5 x}) + 6 e^{0.5 x}

Simplify

2 (2 e^{0.5 x} x - 4 e^{0.5 x}) + 6 e^{0.5 x} : 4 e^{0.5 x} x - 2 e^{0.5 x}

= 4 e^{0.5 x} x - 2 e^{0.5 x}

Add a constant to the solution

= 4 e^{0.5 x} x - 2 e^{0.5 x} + C

f (x) = x^{\frac{2}{7}}

x \to \infty lim (2 x^{2} - 6)

\int x cos (π) d x

x \to 0 lim (\frac{tan ( 2 x )}{0})

\frac{\partial}{\partial x} (\frac{5 x}{4 x ^{2} + 1})

What is the integral of (2x+3)e^{0.5x} ?
The integral of (2x+3)e^{0.5x} is 4e^{0.5x}x-2e^{0.5x}+C