Integrate the following functions-x2-2-3 x33 d x

∫x^2/(2+3x^3)^3dx Let 2+3x^3 =t ⇒ 9x^2 =dt/dx⇒ dx =dt/9x^2 ∴∫x^2/(2+3x^3)^3dx =∫x^2/t^3dt/9x^2=1/9∫1/t^3dt =1/9∫ t^ 3dt =1/9 ( t^ 3+1/ 3+1 )+C =1/9×1/ 2t^ 2+C ...

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Standard XII

Mathematics

Integration by Partial Fractions

Integrate the...

Question

Integrate the following functions.
$\int \frac{x^{2}}{(2 + 3 x^{3})^{3}} d x .$

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Solution

$\int \frac{x^{2}}{(2 + 3 x^{3})^{3}} d x$
Let $2 + 3 x^{3} = t \Rightarrow 9 x^{2} = \frac{d t}{d x} \Rightarrow d x = \frac{d t}{9 x^{2}}$
$∴ \int \frac{x^{2}}{(2 + 3 x^{3})^{3}} d x = \int \frac{x^{2}}{t^{3}} \frac{d t}{9 x^{2}} = \frac{1}{9} \int \frac{1}{t^{3}} d t = \frac{1}{9} \int t^{- 3} d t = \frac{1}{9} (\frac{t^{- 3 + 1}}{- 3 + 1}) + C = \frac{1}{9} \times \frac{1}{- 2} t^{- 2} + C = \frac{- 1}{18 t^{2}} + C = \frac{- 1}{18 (2 + 3 x^{3})^{2}} + C$

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Integrate the following functions. ∫x2(2+3x3)3dx.

∫x2(2+3x3)3dx Let 2+3x3=t⇒9x2=dtdx⇒dx=dt9x2 ∴∫x2(2+3x3)3dx=∫x2t3dt9x2=19∫1t3dt=19∫t−3dt=19(t−3+1−3+1)+C=19×1−2t−2+C=−118t2+C=−118(2+3x3)2+C

Integrate the following functions.
$\int \frac{x^{2}}{(2 + 3 x^{3})^{3}} d x .$