ML Aggarwal Class 12 Maths Solutions Section A Chapter 8 Integrals Ex 8.14

Interactive ML Aggarwal Class 12 ISC Solutions Chapter 8 Integrals Ex 8.14 engage students in active learning and exploration.

ML Aggarwal Class 12 Maths Solutions Section A Chapter 8 Integrals Ex 8.14

Evaluate the following (1 to 5) integrals :

Question 1.
(i) ∫ \(\sqrt{x^2+4 x+6}\) dx (NCERT)
(ii) ∫ \(\sqrt{5-2 x+x^2}\) dx (NCERT Exemplar)
Solution:
(i) Let I = ∫ \(\sqrt{x^2+4 x+6}\) dx

(ii) Let I = ∫ \(\sqrt{5-2 x+x^2}\) dx

Question 1 (old).
(i) ∫ \(\sqrt{x^2+8 x+4}\) dx
(ii) ∫ \(\sqrt{x^2+2 x+5}\) dx (NCERT)
Solution:
(i) I = ∫ \(\sqrt{x^2+8 x+4}\) dx

(ii) Let I = ∫ \(\sqrt{x^2+2 x+5}\) dx

Question 2.
(i) ∫ \(\sqrt{3-2 x-x^2}\) dx (NCERT)
(ii) ∫ \(\sqrt{1+3 x-x^2}\) dx (NCERT)
Solution:
(i) Let I = ∫ \(\sqrt{3-2 x-x^2}\) dx

(ii) Let I = ∫ \(\sqrt{1+3 x-x^2}\) dx

Question 3.
(i) ∫ (x + 3) \(\sqrt{3-4 x-x^2}\) dx
(ii) ∫ (3x + 1) \(\sqrt{4-3 x-2 x^2}\) dx
Solution:
(i) ∫ (x + 3) \(\sqrt{3-4 x-x^2}\) dx

(ii) Let I = ∫ (3x + 1) \(\sqrt{4-3 x-2 x^2}\) dx
= – ∫ – 3 \(\left(x+\frac{1}{3}\right) \sqrt{4-3 x-3 x^2}\) dx

Question 4.
(i) ∫ \(\sqrt{1-4 x-x^2}\) dx (NCERT)
(ii) ∫ \(\sqrt{2 a x-x^2}\) dx (NCERT Exemplar)
Solution:
(i) Let I = ∫ \(\sqrt{1-4 x-x^2}\) dx

(ii) Let I = ∫ \(\sqrt{2 a x-x^2}\) dx

Question 5.
∫ (2x + 3) \(\sqrt{4 x^2+5 x+6}\) dx
Solution:
Let I = ∫ (2x + 3) \(\sqrt{4 x^2+5 x+6}\) dx
Let 2x +3 = A \(\frac{d}{d x}\) (4x² + 5x + 6) + B
= A (8x + 5) + B
∴ 2 = 8A
⇒ A = \(\frac{1}{4}\)
and 5A + B = 3
⇒ B = 3 – \(\frac{5}{4}\) = \(\frac{7}{4}\)