First slide

Methods of integration

Question

$A curve g (x) = \int x^{27} {(1 + x + x^{2})}^{6} (6 x^{2} + 5 x + 4) d x is passing through (0,0) then$

Moderate

A

$g (1) = \frac{3^{7}}{7}$
B

$g (1) = \frac{2^{7}}{7}$
C

$g (2) = \frac{1}{7}$
D

$g (- 1) = \frac{3^{7}}{14}$

Solution

$\begin{array}{l} \begin{aligned} g (x) = \int x^{24} {(1 + x + x^{2})}^{6} (6 x^{5} + 5 x^{4} + 4 x^{3}) d x = \\ \int {(x^{4} + x^{5} + x^{6})}^{6} \cdot (6 x^{5} + 5 x^{4} + 4 x^{3}) d x \\ Put t = x^{6} + x^{5} + x^{4} \Rightarrow \int t^{6} d t = \frac{t^{7}}{7} + C = \frac{{(x^{6} + x^{5} + x^{4})}^{7}}{7} + C \end{aligned} \\ But it passes through (0, 0) \Rightarrow C = 0 \end{array}$

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