?5) Evaluate the following integrals by making an...

Transcript

00:04 Okay, to evaluate this integral, we have to first do the partial fraction decomposition of our 7x minus 2 over the 3x squared minus 2x.

00:16 Hopefully the first thing you notice is the way the denominator factors.

00:20 We can factor an x out of both of those, and so one of the factors is x.

00:25 The other factor is 3x minus 2.

00:28 So those would be the denominators in our partial fraction decomposition.

00:33 So we'll have a over x plus b over the 3x minus 2.

00:39 Now we multiply everything by the common denominator so that we can set up our system of equations to find a and b.

00:47 So multiplying gets rid of the fraction on the left -hand side.

00:51 We just have 7x minus 2.

00:54 Multiplying the right -hand side, each factor will reduce out.

00:58 So we have the numerator times the opposing factor in the other denominator.

01:05 So a times the 3x minus 2 is 3ax minus 2a, and then b times the x is plus bx.

01:16 And now we can equate our terms with each other and equate the coefficients.

01:23 So 7 is going to equal 3a plus b from the x terms, and then the negative 2 has to equal negative 2a from the constant terms.

01:37 Terms.

01:39 So a has to be 1, and then that means that b has to be 4.

01:45 So now we can rewrite our integral as the partial fraction decomposition of 1 over x plus 4 over 3x minus 2 dx, and that integral being evaluated from 1 to 2...

Question

?5) Evaluate the following integrals by making an appropriate U-substitution \begin{equation} \int \frac{x}{(x^2+1)^2} dx \end{equation} 7 pts

Question

?5) Evaluate the following integrals by making an appropriate U-substitution \begin{equation*} \int \frac{x}{(x^2+1)^2} dx \end{equation*} 7 pts

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?5) Evaluate the following integrals by making an appropriate U-substitution \begin{equation} \int \frac{x}{(x^2+1)^2} dx \end{equation} 7 pts