Tamilnadu State Board New Syllabus Samacheer Kalvi 11th Maths Guide Pdf Chapter 11 Integral Calculus Ex 11.11 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.11

Integrate the following with respect to x:

Question 1.
(i) \(\frac{2 x-3}{x^{2}+4 x-12}\)
Answer:
Let 2x – 3 = A \(\frac{\mathrm{d}}{\mathrm{d} x}\) (x2 + 4x – 12) + B
2x – 3 = A (2x + 4) + B
2x – 3 = 2Ax + 4A + B
2A = 2 ⇒ A = 1
4A + B = – 3
4 × 1 + B – 3 ⇒ B = – 3 – 4 = – 7
2x – 3 = 1 (2x + 4) – 7
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.11 1
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.11 2

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.11

(ii) \(\frac{5 x-2}{2+2 x+x^{2}}\)
Answer:
Let 5x – 2 = A \(\frac{\mathrm{d}}{\mathrm{d} x}\) (x2 + 2x + 2) + B
5x – 2 = A (2x + 2) + B
5x – 2 = 2Ax + 2A + B
2A = 5 ⇒ A = \(\frac{5}{2}\)
2A + B = -2
2 × \(\frac{5}{2}\) + B = -2 ⇒ B = -2 – 5 = -7
5x – 2 = \(\frac{5}{2}\) (2x + 2) – 7
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.11 3
Put x2 + 2x + 12 = t
(2x + 2) dx = dt
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.11 4
Put x + 1 = u
dx = du
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.11 5

 

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.10

(iii) \(\frac{3 x+1}{2 x^{2}-2 x+3}\)
Answer:
Let 3x + 1 = A \(\frac{\mathrm{d}}{\mathrm{d} x}\) (2x2 – 2x + 3) + B
3x + 1 = A (4x – 2) + B
3x + 1 = 4Ax – 2A + B
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.11 6
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.11 7
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.11 8

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.10

Question 2.
(i) \(\frac{2 x+1}{\sqrt{9+4 x-x^{2}}}\)
Answer:
Let 2x + 1 = A \(\frac{\mathrm{d}}{\mathrm{d} x}\) (9 + 4x – x2) + B
2x + 1 = A(4 – 2x) + B
2x + 1 = 4A – 2Ax + B
-2A = 2 ⇒ A = -1
4A + B = 1 ⇒ 4 (-1) + B = 1
B = 1 + 4 = 5
B = 5
2x + 1 = – 1 (4 – 2x) + 5
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.11 9
Put 9 + 4x – x2 = t
(4x – 2) dx = dt
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.11 10
Put x – 2 = u
dx = du
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.11 11

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.10

(ii) \(\frac{x+2}{\sqrt{x^{2}-1}}\)
Answer:
Let x + 2 = A \(\frac{\mathrm{d}}{\mathrm{d} x}\) (x2 – 1) + B
x + 2 = A (2x) + B
2A = 1 ⇒ A = \(\frac{1}{2}\)
B = 2
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.11 12
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.11 13

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.10

(iii) \(\frac{2 x+3}{\sqrt{x^{2}+4 x+1}}\)
Answer:
Let 2x + 3 = A \(\frac{\mathrm{d}}{\mathrm{d} x}\) (x2 + 4x + 1) + B
2x + 3 = A (2x + 3) + B
2A = 2 ⇒ A = 1
4A + B = 3
4 × 1 + B = 3 ⇒ B = 3 – 4
B = -1
(2x + 3) = (2x + 4) – 1
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.11 14
Put x2 + 4x + 1 = t
(2x + 4) dx = dt
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.11 15
Put x + 2 = u
dx = dt
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.11 16

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