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) $$\frac{2 x-3}{x^{2}+4 x-12}$$
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

(ii) $$\frac{5 x-2}{2+2 x+x^{2}}$$
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

Put x2 + 2x + 12 = t
(2x + 2) dx = dt

Put x + 1 = u
dx = du

(iii) $$\frac{3 x+1}{2 x^{2}-2 x+3}$$
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

Question 2.
(i) $$\frac{2 x+1}{\sqrt{9+4 x-x^{2}}}$$
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

Put 9 + 4x – x2 = t
(4x – 2) dx = dt

Put x – 2 = u
dx = du

(ii) $$\frac{x+2}{\sqrt{x^{2}-1}}$$
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

(iii) $$\frac{2 x+3}{\sqrt{x^{2}+4 x+1}}$$
Let 2x + 3 = A $$\frac{\mathrm{d}}{\mathrm{d} x}$$ (x2 + 4x + 1) + B