Additional Math – Integration – Integration of a fraction with variable in the numerator and denominator

Integration of a fraction
Image: Solution to an integration question that require differentiation as the first step.

Additional Math Group Tuition – Sunday 4.30pm Class. Integration as a reverse of differentiation

This question is not easy because you need to take a guess for the expression of y.

$latex {Let\ y=\ln (1-2{{x}^{2}})}$

Differentiate y with respect to x.

$latex {\frac{{dy}}{{dx}}=\frac{1}{{1-2{{x}^{2}}}}\times -4x}$

$latex {\frac{{dy}}{{dx}}=\frac{{-4x}}{{1-2{{x}^{2}}}}}$

Integrate both sides of the equation

$latex {y=-4\int{{\frac{x}{{1-2{{x}^{2}}}}}}\ dx}$

$latex {\ln (1-2{{x}^{2}})=-4\int{{\frac{x}{{1-2{{x}^{2}}}}}}\ dx}$

$latex {\int{{\frac{x}{{1-2{{x}^{2}}}}}}\ dx=-\frac{1}{4}\ln (1-2{{x}^{2}})}$



You need to differentiate the y to get an expression for dy/dx, followed by integrating both sides of the expression.

There is no quotient rule for integration, unlike differentiation,so you need to go through the long about way by first differentiating, followed by integrating.

Additional Math (amath) Small Group Tuition on Saturdays in Johor Bahru, Malaysia. SIngapore Syllabus Additonal Math (A Math) Topics.


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