Question

please do the below work and show all steps toward the answer
10. ∫
3𝑥+5
(𝑥−1)(𝑥+2)
𝑑𝑥 11. ∫
5𝑥+7
(𝑥−2)
2(𝑥+1)
𝑑𝑥 12. ∫
𝑥
3+𝑥+4
(𝑥
2+1)(𝑥+2)
𝑑𝑥
13. ∫
𝑥
2−3𝑥+2
𝑥
3+2𝑥+𝑥
𝑑𝑥 14. ∫
𝑥
5+2𝑥
3+4𝑥+4
2𝑥
3+2𝑥
2+𝑥
𝑑𝑥 15. ∫
𝑥+1
(𝑥
2+4𝑥+5)
2 𝑑�

Answer 1

I can do these, but two of the integrals (numbers 13 and 14) are ambiguous from the formatting. I’ll solve 10, 11, 12 and 15 in full detail below, and then ask you to clarify 13 and 14 (showing the likely interpretations).

10) ∫ (3x + 5)/((x − 1)(x + 2)) dx

Use partial fractions:
(3x + 5)/((x − 1)(x + 2)) = A/(x − 1) + B/(x + 2).

Multiply through: 3x + 5 = A(x + 2) + B(x − 1) = (A + B)x + (2A − B).

Compare coefficients:
A + B = 3
2A − B = 5

Solve: add the two equations: 3A = 8 ⇒ A = 8/3. Then B = 3 − A = 3 − 8/3 = 1/3.

So the integral is
∫ (8/3)/(x − 1) dx + ∫ (1/3)/(x + 2) dx
= (8/3) ln|x − 1| + (1/3) ln|x + 2| + C.

11) ∫ (5x + 7)/((x − 2)^2 (x + 1)) dx

Use partial fractions with a repeated factor:
(5x + 7)/((x − 2)^2 (x + 1)) = A/(x − 2) + B/(x − 2)^2 + C/(x + 1).

Multiply through:
5x + 7 = A(x − 2)(x + 1) + B(x + 1) + C(x − 2)^2.

Expand and collect coefficients. One convenient way is to expand:
A(x^2 − x − 2) + B(x + 1) + C(x^2 − 4x + 4)
= (A + C)x^2 + (−A − 4C + B)x + (−2A + B + 4C).

Because left side has no x^2 term, A + C = 0 ⇒ C = −A.

Compare x-coeff and constant:
x-coeff: 3A + B = 5 (since −A −4C + B = −A −4(−A) + B = 3A + B)
constant: B − 6A = 7 (since −2A + B + 4C = −2A + B + 4(−A) = B −6A)

Solve: from 3A + B = 5 and B − 6A = 7. Subtract the second from the first:
(3A + B) − (B − 6A) = 5 − 7 ⇒ 9A = −2 ⇒ A = −2/9.
Then C = −A = 2/9. B = 5 − 3A = 5 − 3(−2/9) = 5 + 6/9 = 5 + 2/3 = 17/3.

Therefore
∫(5x + 7)/((x − 2)^2 (x + 1)) dx
= ∫[A/(x − 2) + B/(x − 2)^2 + C/(x + 1)] dx
= A ln|x − 2| − B/(x − 2) + C ln|x + 1| + C1

Substitute constants:
= (−2/9) ln|x − 2| − (17/3)(1/(x − 2)) + (2/9) ln|x + 1| + C.

(You can combine the logarithm terms if desired.)

12) ∫ (x^3 + x + 4)/((x^2 + 1)(x + 2)) dx

First do polynomial division because degree numerator = degree denominator:
Denominator expanded: (x^2 + 1)(x + 2) = x^3 + 2x^2 + x + 2.
Divide x^3 + 0x^2 + x + 4 by x^3 + 2x^2 + x + 2:

Quotient = 1, remainder = (x^3 + x + 4) − (x^3 + 2x^2 + x + 2) = −2x^2 + 0x + 2.

So integrand = 1 + (−2x^2 + 2)/((x^2 + 1)(x + 2)).

Do partial fractions for the proper fraction:
(−2x^2 + 2)/((x^2 + 1)(x + 2)) = (Ax + B)/(x^2 + 1) + C/(x + 2).

Multiply through:
−2x^2 + 2 = (Ax + B)(x + 2) + C(x^2 + 1)
= (A + C)x^2 + (2A + B)x + (2B + C).

Match coefficients:
A + C = −2
2A + B = 0
2B + C = 2

From 2A + B = 0 ⇒ B = −2A.
Then 2(−2A) + C = 2 ⇒ −4A + C = 2.
Also A + C = −2 ⇒ C = −2 − A.
Substitute: −4A + (−2 − A) = 2 ⇒ −5A − 2 = 2 ⇒ −5A = 4 ⇒ A = −4/5.
Then B = −2A = 8/5. C = −2 − A = −2 + 4/5 = −6/5.

Thus integrand = 1 + (A x + B)/(x^2 + 1) + C/(x + 2).

Integrate termwise:
∫1 dx = x.
∫ A x/(x^2 + 1) dx = (A/2) ln(x^2 + 1).
∫ B/(x^2 + 1) dx = B arctan x.
∫ C/(x + 2) dx = C ln|x + 2|.

Substitute the constants A = −4/5, B = 8/5, C = −6/5:
Integral = x + (−4/5)/2 · ln(x^2 + 1) + (8/5) arctan x + (−6/5) ln|x + 2| + C
= x − (2/5) ln(x^2 + 1) + (8/5) arctan x − (6/5) ln|x + 2| + C.

15) ∫ (x + 1)/(x^2 + 4x + 5)^2 dx

Complete the square in the denominator: x^2 + 4x + 5 = (x + 2)^2 + 1. Let u = x + 2 ⇒ du = dx and x + 1 = u − 1.

Integral becomes ∫ (u − 1)/(u^2 + 1)^2 du = ∫ u/(u^2 + 1)^2 du − ∫ 1/(u^2 + 1)^2 du.

First term: let t = u^2 + 1 ⇒ dt = 2u du ⇒ u du = dt/2, so
∫ u/(u^2 + 1)^2 du = (1/2) ∫ dt/t^2 = (1/2)(−1/t) = −1/(2(u^2 + 1)).

Second term: use the standard integral
∫ du/(u^2 + 1)^2 = u/(2(u^2 + 1)) + (1/2) arctan u + C.

Therefore the whole integral is
−1/(2(u^2 + 1)) − [u/(2(u^2 + 1)) + (1/2) arctan u] + C
= −(u + 1)/(2(u^2 + 1)) − (1/2) arctan u + C.

Replace u = x + 2 and u^2 + 1 = x^2 + 4x + 5:
Result = −(x + 3)/(2(x^2 + 4x + 5)) − (1/2) arctan(x + 2) + C.

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Please confirm 13 and 14 — the text you gave is ambiguous. For example:

- For 13 you wrote "∫ x 2−3x+2 x 3+2x+ x dx". Possible intended integrands include
• (x^2 − 3x + 2)/(x^3 + 2x + 1) or
• (x^2 − 3x + 2)/(x^3 + 2x + x) = (x^2 − 3x + 2)/(x^3 + 3x) = (x^2 − 3x + 2)/(x(x^2 + 3)).
Please tell me which denominator (exactly), with parentheses and powers if needed.

- For 14 the expression is also unclear. Please retype 14 exactly using ^ for powers and parentheses where needed (for instance: (x^5 + 2x^3 + ...)/((3 + 4x + 4x^2)(2x^3 + 2x^2 + x)) or whatever the intended factors are).

If you confirm the exact forms for 13 and 14 I will continue and show all steps for them as well.