₀^{4}(x+e²x})dx - NCERT Class XII Mathematics Chapter -

NCERT Class XII Mathematics Chapter - - Solutions

Exercise - 7.8
Evaluate the following definite integrals as limit of sums.

Question : 174 of 271

Marks: +1, -0

\int\limits_{0}^{4}(x+e^{2x})dx

Solution:

Let

f(x) = x + e^{2x}

\Rightarrow \int\limits_{0}^{4} f(x) dx = \int\limits_{0}^{4}(x + e^{2x})dx

a = 0, b = 4

and

nh = b - a = 4 - 0 = 4

Now,

\int\limits_{a}^{b} f(x) = \lim\limits_{h \to 0} h[f(a) + f(a+h) + f(a+2h) + \dots + f(a+(n-1)h)]

= \lim\limits_{h \to 0} h[f(0) + f(h) + f(2h) + \dots + f((n-1)h)]

= \lim\limits_{h \to 0} h[(0+e^0) + (h+e^{2h}) + (2h+e^{4h}) + \dots + ((n-1)h + e^{2(n-1)h})]

= \lim\limits_{h \to 0} h\left[ h \cdot \frac{(n-1)(1+(n-1))}{2} + e^{2h} \cdot \frac{1 \cdot (e^{2nh}-1)}{e^{2h}-1} \right]

= \frac{4 \times 4}{2} + \frac{1}{2} \cdot \frac{e^0}{\log e}(e^8 - 1)

= \frac{e^8+15}{2}

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