What is the integral from 4 to 8 of (2x+15)e^{0.25x} ?

The integral from 4 to 8 of (2x+15)e^{0.25x} is 516.69625…

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integral from 4 to 8 of (2x+15)e^{0.25x}

\int_{4}^{8} (2 x + 15) e^{0.25 x} d x

516.69625 \dots

Solution steps

\int_{4}^{8} (2 x + 15) e^{0.25 x} d x

Multiply by the conjugate of

2 x + 15 : \frac{4 x ^{2} - 225}{2 x - 15}

= \int_{4}^{8} \frac{4 x ^{2} - 225}{2 x - 15} e^{0.25 x} d x

Find undefined (singularity) points:

x = \frac{15}{2}

If exist

b, a < b < c, f (b) =

undefined

, \int_{a}^{c} f (x) d x = \int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x

= \int_{4}^{\frac{15}{2}} \frac{4 x ^{2} - 225}{2 x - 15} e^{0.25 x} d x + \int_{\frac{15}{2}}^{8} \frac{4 x ^{2} - 225}{2 x - 15} e^{0.25 x} d x

\int_{4}^{\frac{15}{2}} \frac{4 x ^{2} - 225}{2 x - 15} e^{0.25 x} d x = 410.73517 \dots

\int_{\frac{15}{2}}^{8} \frac{4 x ^{2} - 225}{2 x - 15} e^{0.25 x} d x = 105.96107 \dots

= 410.73517 \dots + 105.96107 \dots

Simplify

= 516.69625 \dots

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What is the integral from 4 to 8 of (2x+15)e^{0.25x} ?
The integral from 4 to 8 of (2x+15)e^{0.25x} is 516.69625…