Tamilnadu State Board New Syllabus Samacheer Kalvi 11th Maths Guide Pdf Chapter 11 கணங்கள், தொடர்புகள் மற்றும் naசார்புகள் Ex 11.1 Textbook Questions and Answers, Notes.
TN Board 11th Maths Solutions Chapter 11 கணங்கள், தொடர்புகள் மற்றும் சார்புகள் Ex 11.1
கீழ்க்காண்பனவற்றைத் தொகையிடுக :
Question 1.
(i) x11
(ii) \(\frac{1}{x^{7}}\)
(iii) \(\sqrt[3]{x^{4}}\)
(iv) \(\left(x^{5}\right)^{\frac{1}{8}}\)
(i) x11
தீர்வு :
∫ xn dx = \(\frac{x^{n+1}}{n+1}\) + c
∴ ∫ x11 dx = \(\frac{x^{11+1}}{11+1}\) + c
= \(\frac{x^{12}}{12}\) + c
(ii) ∫ \(\frac{1}{x^{7}}\) dx = ∫x-7 dx
= \(\frac{x^{-7+1}}{-7+1}\)
= \(\frac{x^{-6}}{-6}\) + c
= \(-\frac{1}{6 x^{6}}\) + c
(iii) ∫ \(\sqrt[3]{x^{4}}\) dx = ∫ (x4)\(\frac{1}{3}\) dx
= x\(\frac{4}{3}\) dx
= \(\frac{x^{\frac{4}{3}}+1}{\frac{4}{3}+1}\) + c
= \(\frac{x^{\frac{7}{3}}}{\frac{7}{3}}\) + c
= \(\frac{3}{7} x^{\frac{7}{3}}\) + c
(iv) ∫ \(\left(x^{5}\right)^{\frac{1}{8}}\) dx
= ∫ x\(\frac{5}{8}\) dx
= \(\frac{x^{\frac{5}{8}+1}}{\frac{5}{8}+1}\)+ c
= \(\frac{x^{\frac{13}{8}}}{\frac{13}{8}}\) + c
= \(\frac{8}{13} x^{\frac{13}{8}}\) + c
Question 2.
(i) \(\frac{1}{\sin ^{2} x}\)
(ii) \(\frac{\tan x}{\cos x}\)
(iii) \(\frac{\cos x}{\sin ^{2} x}\)
(iv) \(\frac{1}{\cos ^{2} x}\)
தீர்வு :
\(\frac{1}{\sin ^{2} x}\)
∫ \(\frac{1}{\sin ^{2} x}\) dx = \(\frac{1}{\sin ^{2} x}\) cosec2 dx = – cot x + c
(ii) \(\frac{\tan x}{\cos x}\)
∫ \(\frac{\tan x}{\cos x}\) dx = ∫ tan x × \(\frac{1}{\cos x}\) dx
= ∫ tan x sec x dx = sec x + c
(iii) \(\frac{\cos x}{\sin ^{2} x}\)
∫ \(\frac{\cos x}{\sin ^{2} x}\) dx = ∫ \(\frac{\cos x}{\sin x} \times \frac{1}{\sin x}\) dx
= ∫ cot x cosec x dx = – cosec x + c
(iv) \(\frac{1}{\cos ^{2} x}\)
∫ \(\frac{1}{\cos ^{2} x}\) dx = ∫ sec2 dx = tan x + c
Question 3.
(i) 123
(ii) \(\frac{x^{24}}{x^{25}}\)
(iii) ex
தீர்வு :
(i) 123 = ∫ 123 dx = 123 ∫ dx = 123 + c
(ii) \(\frac{x^{24}}{x^{25}}\)
= ∫ \(\frac{x^{24}}{x^{25}}\) dx = ∫ \(\frac{1}{x}\) dx
= log |x| + c
(iii) ex
= ∫ ex dx = ex + c
Question 4.
(i) (1 + x2)-1
(ii) (1 – x2)\(-\frac{1}{2}\)
தீர்வு :
(i) (1 + x2)-1
= ∫ (1 + x2)-1 dx
= ∫ \(\frac{1}{1+x^{2}}\) dx = tan-1 x + c
(ii) ∫ (1 – x2)\(-\frac{1}{2}\) dx
= ∫ \(\frac{1}{\left(1-x^{2}\right)^{\frac{1}{2}}}\) dx
= ∫ \(\frac{d x}{\sqrt{1-x^{2}}}\) + cOS
= sin-1 + c