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Integrate each of the given functions.$$\int \frac{10 x^{3}+40 x^{2}+22 x+7}{\left(4 x^{2}+1\right)\left(x^{2}+6 x+10\right)} d x$$

$\frac{1}{4} \ln \left(4 x^{2}+1\right)+\ln \left|x^{2}+6 x+10\right|+\tan ^{-1}(x+3)+C$

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 10

Integration by Partial Fractions: Other Cases

Integrals

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in the problem we have been given integration 10 X Q plus 40 X square Plus 22 x plus seven upon four x squared plus one into X square plus six X less than the X. Now this is equal to integration two x upon four x Squire plus one. Class two x plus seven upon excess choir plus 66 plus 10 dx. This is equal to twice Integration X and dx Upon four x square plus one plus digression, two x plus seven upon X squared plus six six plus then dicks. Now for first part we put four X Squire plus one as indeed And it is eight x. The ecstatic was to visit now further this is written as two upon eight. Integration digit upon jade That equals to run upon four Ellen mud four x squared plus one plus decoration two x plus six lb X squared plus six X plus 10 D. X plus the X upon X squared plus six six plus 10. So further this equals two, one of 14 11 model four X square plus one plus integration dessert upon jude. As we have assumed, X squared plus six X plus 10. That equals two injured. Hence we have two X. Dx. Or that is two X plus six two x plus six. The x equals two digit. That's why we are writing it. Develop on jade for this. Plus integration went upon X plus three. Holy square plus one. Now this is it indigenous one upon 4. Ellen Mod four x square plus one plus Ellen model X square Plus six x plus 10 plus 10. Universe X plus three plus C. So this is the answer to the problem.

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