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The integral $\displaystyle\int\dfrac{2x^{12}+5x^9}{(x^5+x^3+1)^3}\,dx$ equals (where $C$ is an arbitrary constant):
Divide numerator and denominator by $x^{15}$:
$\displaystyle\int\dfrac{2x^{-3}+5x^{-6}}{(1+x^{-2}+x^{-5})^3}\,dx$
Let $t=1+x^{-2}+x^{-5}$. Then $dt=(-2x^{-3}-5x^{-6})dx$, so the numerator (divided by $x^{15}$) gives $-dt$:
$=-\displaystyle\int\dfrac{dt}{t^3} \cdot \dfrac{1}{x^{15}/x^{15}}$
Wait — let me redo with $t = x^{-5}+x^{-3}+1$... Actually substituting directly: $t=1+x^{-2}+x^{-5}$, $-dt=(2x^{-3}+5x^{-6})dx$.
Integral $= -\int t^{-3}dt = \dfrac{1}{2t^2}+C = \dfrac{x^{10}}{2(x^5+x^3+1)^2}+C$
The 18th Amendment to the Constitution of Pakistan (2010) primarily focused on:
The 18th Amendment significantly increased provincial autonomy and removed the President's power to dissolve Parliament.
During isothermal expansion of an ideal gas:
Isothermal: ΔT=0, so internal energy constant (ideal gas). Gas does work on surroundings.
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