Evaluate - 13x2-3x-exdx- as the limit of the sum-

Let I = ∫_1^3 x^2+3x+e^xdx ⇒ I = #160; ∫_1^3 x^2dx+3 ∫_1^3xdx+ ∫_1^3e^xdx = x^3/3 |_1^3 +3x^2/2 |_1^3+e^x |_1^3 = 27 1/3+3/2(9 1)+e^3 e I ...

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Standard XII

Mathematics

Property 7

Evaluate ∫ ...

Question

Evaluate $\int_{1}^{3} (x^{2} + 3 x + e^{x}) d x$ , as the limit of the sum.

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Solution

Let $I = \int_{1}^{3} x^{2} + 3 x + e^{x} d x$
$\Rightarrow I = \int_{1}^{3} x^{2} d x + 3 \int_{1}^{3} x d x + \int_{1}^{3} e^{x} d x$
$= \frac{x^{3}}{3} |_{1}^{3} + 3 \frac{x^{2}}{2} |_{1}^{3} + e^{x} |_{1}^{3} = \frac{27 - 1}{3} + \frac{3}{2} (9 - 1) + e^{3} - e$
$I = \frac{26}{3} + 12 + e^{3} - e$

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Evaluate ∫31(x2+3x+ex)dx, as the limit of the sum.

Let I=∫31x2+3x+exdx⇒I=∫31x2dx+3∫31xdx+∫31exdx=x33|31+3x22|31+ex|31=27−13+32(9−1)+e3−eI=263+12+e3−e

Evaluate $\int_{1}^{3} (x^{2} + 3 x + e^{x}) d x$ , as the limit of the sum.