Reverse Chain Rule ?

Integration Using The Substitution Rule
Integration Using The Substitution Rule

Time was wasted when the lecturer integrates via substitution with respect to x:

[maths]\frac{x^{3}}{\sqrt{x^{4}+y^{2}}}[/maths]

which is also:

[maths]{x^{3}}.(x^{4}+y^{2})^{-\frac{1}{2}}[/maths]

Then integrating via inspection to get:

[maths]2.\frac{1}{4}.(x^{4}+y^{2})^{\frac{1}{2}}+C[/maths]

2 is to “cancel out” the constant term after differentiating the inside chain rule w.r.t. x

1/4 is to “cancel out” the power which comes down when you pretend to differentiate the integral again.

[maths]\frac{1}{2}.(x^{4}+y^{2})^{\frac{1}{2}}+C[/maths]

If you have a bit more time differentiate the integral w.r.t. x to check.

Simple.

See the pattern?

Also makes the quotient rule obsolete, which I forgot.

The quotient rule is just a derivation of the product rule with the chain rule.

Jeez, it doesn’t waste that much time + you’ve still said you lost a few marks.
But I guess if you’re completely pro at it, reverse chain rule is better time-wise, if you’re 100% confident you can do it without making a mistake.
Also that thread is nice but you don’t have to do integration by parts in ANY of them ==;;

Time wasted?

How about exams?

Have you even bothered to check this thread?

http://community.boredofstudies.org/12/mathematics/179888/challenge.html\

Try solving them all in under 10 minutes using either method (substitution or inspection).

I agree with Andrew. For someone like me who makes tonnes of silly errors, doing it by sub is the easiest way.

Because you haven’t worked on enough integrals and/or your brain isn’t capable enough to solve by inspection.

With experience which I gained from solving countless integrals, integration via substitution will become obsolete for ‘reverse chain rule integrals’.

I disagree. I think you are more likely to make silly mistakes in subsutition than in reverse chain rule (provided you are effective at either). In a formal substitution you have to remember to substitute back into original variable or change limits. Forgetting to do this is a common silly mistake. You don’t have to worry about ANY of these in the reverse chain rule.

As he said, especially with the limits.

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