First slide

Methods of integration

Question

$\int 3 x^{2} 3^{\log (x^{3} + 4)} d x =____$

Moderate

A

$\frac{3^{2 \log (x^{3} + 4)}}{2} + C$
B

$3^{\log (x^{3} + 4)} + C$
C

$(x^{3} + 4)$
D

$\frac{{(x^{3} + 4)}^{1 + \log 3}}{1 + \log 3} + C$

Solution

$\begin{array}{l} I = \int 3 x^{2} 3^{\log (x^{3} + 4)} d x \\ Let x^{3} + 4 = t \Rightarrow 3 x^{2} d x = d t \\ I = \int 3^{\log t} d t \\ = \int t^{\log 3} d t (∵ a^{\log x^{b}} = b^{\log x^{a}}) \\ = \frac{t^{\log 3 + 1}}{1 + \log 3} + C = \frac{{(x^{3} + 4)}^{1 + \log 3}}{1 + \log 3} + C \end{array}$

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