Evaluate the integrals in Exercises 69–134. The integrals are ...

Question

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ (z + 1) / [z²(z² + 4)] dz

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Evaluate the integral. The integral of x minus 2 divided by x 2 multiplied by parentheses of x 2 + 9 dx. OK. So it appears for this particular prompt for us to simply evaluate the integral that is provided to us. So now that we know that we're ultimately trying to evaluate this integral, our first step that we need to take is we need to recall and use partial fraction decomposition. So we will need to set up the decomposition for our integrand, which is X minus 2. Divided by x 2 multiplied by parentheses of x 2 + 9, which will be equal to a divided by x plus b divided by x 2 plus c x + d. Divided by X2 + 9, or specifically C multiplied by X + D divided by X2 + 9. So now, what we need to do is we need to multiply both sides by X2, multiply by parentheses of X2 + 9. And let's make a note of this. So we're multiplying both sides by X 2 multiplied by parentheses of X2 + 9. And when we multiply both sides by this expression. We will be able to write x minus 2 is equal to ax multiplied by parentheses of x 2 + 9. Plus b multiplied by x 2 + 9 in parentheses plus parentheses of c multiplied by x + d multiplied by x to the power of 2. OK, and this will simplify to x minus 2 is equal to parentheses of. A plus C multiplied by, so it's gonna be A plus C in parentheses multiplied by X 3 + B + D multiplied by X 2, plus 9 multiplied by A multiplied by X plus 9 B. OK. So now we need to equate the coefficients of like powers of X, and for the X to the power of 3 term, we need to have A + C equal to 0. for X2 term, we need to have B + D. is equal to 0, and for our X term, we need to have 9 multiplied by A. 9A is equal to 1, which will mean that A means when we rearrange our expression to isolate sulfur A, you'll find that A is equal to 1 divided by 9. OK. So with that said, we now need to solve for our constant term, and our constant term is 9 multiplied by b is equal to -2, which will mean that b by itself is equal to -2 divided by 9. Fantastic. And now we need to solve for C and D, and C is equal to A, which is equal to -1 divided by 9. And D. Is equal to B, which is equal to 2 divided by 9. So now we need to substitute our new variables back into our decomposition. So we need to write that x minus 2 divided by x 2 multiplied by parentheses x 2 + 9 is equal to 1 divided by 9 x minus 2 multiplied by 9 x 2. Plus 19 x + 2/9 Divided by X2 + 9. Which shall be all equal to 1/9 multiplied by square brackets of 1 divided by x minus 2 divided by x 2 minus x divided by x 2 + 9 + 2 divided by x 2 + 9. Fantastic. So now at this stage for our second step is we need to begin our integration. So now we will need to integrate term by term. Meaning we will need to take the integral of X minus 2. Divided by x 2 multiplied by parentheses of x 2 + 9. D x, which will be equal to 1/9 multiplied by the integral of 1 divided by x dx minus 2 divided by 9 or 2/9 multiplied by the integral of x to the power of -2 d x minus 1/9 multiplied by the integral of x divided by x 2 + 9 d x. Plus 2/9 multiplied by the integral of 1 divided by x 2 + 9 DX. OK. We are on a roll here. So that will mean that we can when we solve for each of our integral individually, we will find that 1/9 multiplied by the integral of 1 divided by x, the x will be equal to 1/9 multiplied by the natural log of the absolute value of x. Plus C subscript 1 where c denotes some constant variable for our second integral -2/9 multiplied by the integral of x to the power of -2 d x will be equal to, which when we write our final answer in its most simplified form, it will be equal to drum roll. 2 divided by 9 x plus C subscript 2. And for our 3rd integral, which is the integral. Of X divided by X2 + 9 DX, we will need to use our substitution method, so we need to state that you will be equal to X2 + 9. Which will mean that DU. By DX, so DU by DX will be equal to 2X. Which will mean that 1/2 DU will be equal to XDX. So that will mean that -1/9 multiplied by the integral of x divided by x 2 + 9 dx will be equal to 1/18 multiplied by the integral of 1 divided by UDU. Which will be equal to 1/18 multiplied by the natural log of the absolute value of u. Plus see subscript 3. Which when we simplify this down to its most simple form, we will find that it's equal to -1 divided by 18, so 1 divided by 18, so 1/18. Multiplied by the natural log of parentheses of x to the 2 + 9 + c subscript 3, and we can remove the absolute value signs because anything inside of the absolute value signs will be positive. So With that said, solving for our fourth integral, which is 2/9 multiplied by the integral of 1 divided by x 2, plus 3 of 2, the XP is once again 32 is 9, which will be simply equal to 2 divided by 27 multiplied by inverse tangent of x divided by 3 plus C subscript 4. OK, so our third and final step is we need to solve for our final result by combining all terms. So we need to take the integral of X minus 2 divided by x 2 multiplied by parentheses of x 2 + 9. The x, which will be equal to 1/9 multiplied by the natural log of the absolute value of x plus 2 divided by 9 x minus 1/18 multiplied by the natural log of the absolute value of x 2 + 9. Plus 2 divided by 27 multiplied by inverse tangent of x divided by 3 plus C, noting that C will be equal to c1 plus C2 plus C3. Plus C4, fantastic. So now we need to note that we can simplify this expression into a more compact form, and when we simplify it down to its most simplified compact form, we will find that it's equal to. 1/18, 1 divided by 18 multiplied by the natural log of parentheses of x to the power of 2 divided by x 2 plus 9. Plus 2 divided by 9 x plus 2 divided by 27 multiplied by inverse tangent of x divided by 3 plus c and that's it. That is our final answer. Hooray, we did it. So with that in mind, we can now go back up to the top to recap what we've learned. So based on what we've discovered together, we could safely say that once again our final answer for our evaluated integral will be once again drum roll. 1 divided by 18 multiplied by the natural log of x to the power of 2 divided by x 2 + 9 + 2 divided by 9 x plus 2 divided by 27 multiplied by inverse tangent of x divided by 3 plus c. And that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.∫ (z + 1) / [z²(z² + 4)] dz

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Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ (z + 1) / [z²(z² + 4)] dz