Integrate the function: \(\frac{5x-2}{1+2x+3x^2}\)
Solution and Explanation
Let 5x-2 = A \(\frac{d}{dx}\)(1+2x+3x2)+B
⇒ 5x - 2 = A(2+6x)+B
Equating the coefficient of x and constant term on both sides, we obtain
5 = 6A ⇒ A\(\frac{5}{6}\)
2A+B = -2 ⇒ B= -11/3
∴ 5x-2 = 5/6(2+6x)+(-11/3)
⇒ ∫\(\frac{5x-2}{1+2x+3x^2}\)dx = ∫5/6(2+6x)-11/3/1+2x+3x2 dx
=5/6 ∫2+6x/1+2x+3x2 dx-11/3 ∫1/1+2x+3x2 dx
Let I1 = ∫2+6x/1+2x+3x2 dx and I2 = ∫1/1+2x+3x2 dx
∴ ∫\(\frac{5x-2}{1+2x+3x^2}\) dx = 5/6I1-11/3I2 ...(1)
I1 = ∫2+6x/1+2x+3x dx
Let 1+2x+3x2 = t
⇒ (2+6x)dx = dt
∴ I1 = ∫dt/t
I1 = log|t|
I1 = log|1+2x+3x2| ...(2)
I2 = ∫1/1+2x+3x2 dx
1+2x+3x2 can be written as 1+3(x2+2/3x).
Therefore,
1+3(x2+2/3x)
=1+3(x2+2/3x+1/9-1/9)
=1+3(x+1/3)2-1/3
=2/3+3(x+1/3)2
=3[(x+1/3)2+2/9]
=3[(x+1/3)2+(√2/3)2]
I2 = 1/3 ∫1/[(x+1/3)2+(√2/3)2]dx
=1/3[1/√2/3tan-1(3x+1)/√2)]
=1/√2tan-1(3x+1/√2) ...(3)
Substituting equations (2) and (3) in equation (1), we obtain
∫5x-2/1+2x+3x2 dx = 5/6[log|1+2x+3x2|]-11/3[1/√2tan-1(3x+1/√2)]+C
=5/6log|1+2x+3x2|-11/3√2tan-1(3x+1/√2)+C
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Questions Asked in CBSE CLASS XII exam
- Altima Ltd. invited applications for 2,00,000 equity shares of ₹ 10 at a premium of ₹ 4 per share. Amount payable:
On application and allotment – ₹ 7 (incl. ₹ 1 premium)
On first and final call – Balance.
Applications received for 2,40,000 shares. 30,000 rejected. Manvi allotted 4,000 shares failed to pay first and final call. Her shares were forfeited. These were reissued at ₹ 4 per share fully paid-up.
Pass journal entries in the books of Altima Ltd.- CBSE CLASS XII - 2025
- Accounting for Share Capital
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"There is widely spatial variation in different sectors of work participation in India." Evaluate the statement with suitable examples.
- CBSE CLASS XII - 2025
- Population
Alexia Limited invited applications for issuing 1,00,000 equity shares of ₹ 10 each at premium of ₹ 10 per share.
The amount was payable as follows:- On application ₹ 9 per share (Including premium ₹ 6 per share)
- On allotment ₹ 8 per share (Including premium ₹ 4 per share)
- On first and final call ₹ 3 per share.
Applications were received for 1,50,000 equity shares and allotment was made to the applicants as follows:
Category A: Applicants for 90,000 shares were allotted 70,000 shares.
Category B: Applicants for 60,000 shares were allotted 30,000 shares.
Excess money received on application was adjusted towards allotment and first and final call.
Shekhar, who had applied for 1200 shares failed to pay the first and final call. Shekhar belonged to category B.
Pass necessary journal entries for the above transactions in the books of Alexia Limited. Open calls in arrears and calls in advance account, wherever necessary.- Shweta mixed two liquids A and B of 10 mL each. After mixing, the volume of the solution was found to be 20.2 mL.
(i) Why was there a volume change after mixing the liquids?
(ii) Will there be an increase or decrease of temperature after mixing?
(iii) Give one example for this type of solution.- CBSE CLASS XII - 2025
- Solutions
Concepts Used:
Integrals of Some Particular Functions
There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.
Integrals of Some Particular Functions:
- ∫1/(x2 – a2) dx = (1/2a) log|(x – a)/(x + a)| + C
- ∫1/(a2 – x2) dx = (1/2a) log|(a + x)/(a – x)| + C
- ∫1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
- ∫1/√(x2 – a2) dx = log|x + √(x2 – a2)| + C
- ∫1/√(a2 – x2) dx = sin-1(x/a) + C
- ∫1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C
These are tabulated below along with the meaning of each part.
