Integration of (3x(x^2+3)^4) dx

Step-by-step explanation:

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It is apparently obvious we could expand the bracket and integrate term-by-term. This method would work but is very time consuming (and you could easily make a mistake) so we use a different method: integration by substitution.

Integration by substitution involves swapping the variable for another variable which depends on x: . (We are going to choose for this question).

The very first step is to choose a suitable substitution. That is, an equation which is going to make the integration easier. There is a trick for spotting this however: if an integral contains both a term and it's derivative then use the substitution .

Your integral contains the term . The derivative is and (ignoring the constants) we see is also in the integral and so the substitution will unravel this integral!

Step 2: We must now swap the variable of integration from x to u. That means interchanging all the x's in the integrand (the term being integrated) for u's and also swapping (dx" to "du").

Then,

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The substitution has made this integral is easy to solve!

Finally we can substitute back to get the answer in terms of x:

where's the diagram?

step-by-step explanation: