OP Malhotra Class 12 Maths Solutions Chapter 13 Indefinite Integral-1 Ex 13(b)

Interactive ISC Class 12 Maths Solutions OP Malhotra Chapter 13 Indefinite Integral-1 Ex 13(b) engage students in active learning and exploration.

S Chand Class 12 ICSE Maths Solutions Chapter 13 Indefinite Integral-1 Ex 13(b)

Question 1.
\(\int \frac{\left(x^2-1\right)^2}{x^3} d x\)
Solution:

Question 2.
\(\int \frac{x+1}{x^2-1} d x\)
Solution:

Question 3.
\(\int e^{-x} d x\)
Solution:

Question 4.
\(\int e^{3 x} d x\)
Solution:
\(\int e^{3 x} d x=\frac{e^{3 x}}{3}+\mathrm{C}\)

Question 5.
\(\int a^{2 x} d x\)
Solution:

Question 6.
\(\int\left(e^{3 a \log _e x}+e^{3 x \log _e a}\right) d x\)
Solution:
Let I = \(\int\left(e^{3 a \log _e x}+e^{3 x \log _e a}\right) d x\)
= \(\int\left(e^{\log _e x^{3 a}}+e^{\log _e a^{3 x}}\right) d x\)
= \(\int x^{3 a} d x+\int a^{3 x} d x\) [∵ e^logx = x]
= \(\frac{x^{3 a+1}}{3 a+1}\) + \(\frac{a^{3 x}}{3 \log a}\) + C

Question 7.
\(\int \frac{3 e^{2 x}+3 e^{4 x}}{e^x+e^{-x}} d x\)
Solution:
\(\int \frac{3 e^{2 x}+3 e^{4 x}}{e^x+e^{-x}} d x\)
\(=\int \frac{3 e^{2 x}\left(1+e^{2 x}\right) e^x}{e^{2 x}+1} d x=\int 3 e^{3 x} d x\)
\(=3 \frac{e^{3 x}}{3}+\mathrm{C}=e^{3 x}+\mathrm{C}\)

Question 8.
\(\int\left(\frac{5 x+7}{x}+e^x\right) d x\)
Solution:
\(\int\left(\frac{5 x+7}{x}+e^x\right) d x\)
\(=\int\left[5+\frac{7}{x}+e^x\right] d x\)
\(=\int 5 d x+7 \int \frac{1}{x} d x+\int e^x d x\)
\(=5 x+7 \log |x|+e^x+C\)

Question 9.
\(\int \frac{a x^2+b x+c}{x^2} d x\)
Solution:
Let I = \(\int \frac{a x^2+b x+c}{x^2} d x\)
= \(\int\left[a+\frac{b}{x}+\frac{c}{x^2}\right] d x\)
= \(a x+b \log |x|-\frac{\mathrm{c}}{x}+\mathrm{C}^{\prime}\)

Question 10.
\(\int \frac{d x}{\sqrt{16-x^2}}\)
Solution:
\(\int \frac{d x}{\sqrt{16-x^2}}\) = \(\int \frac{d x}{\sqrt{4^2-x^2}}\)
= \(\sin ^{-1} \frac{x}{4}+C\)

Question 11.
\(\int \frac{d x}{\sqrt{25-4 x^2}}\)
Solution:

Question 12.
\(\int \frac{d x}{4+x^2}\)
Solution:
\(\int \frac{d x}{4+x^2}\) = \(\int \frac{d x}{2^2+x^2}\)
= \(\frac { 1 }{ 2 }\) tan¹\(\left(\frac{x}{2}\right)+C\)

Question 13.
\(\int \frac{d x}{16+9 x^2}\)
Solution:

Question 14.

Question 15.
\(\int\left(x+\frac{1}{x}\right)\left(x^2+\frac{1}{x^2}\right) d x\)
Solution:
\(\int\left(x+\frac{1}{x}\right)\left(x^2+\frac{1}{x^2}\right) d x\)
\(=\int\left(x^3+\frac{1}{x}+x+\frac{1}{x^3}\right) d x\)
\(=\frac{x^4}{4}+\log |x|+\frac{x^2}{2}+\frac{x^{-3+1}}{-3+1}+C\)
\(=\frac{x^4}{4}+\log |x|+\frac{x^2}{2}-\frac{1}{2 x^2}+C\)

Question 16.
\(\int \frac{(x+2)\left(4 x^2-5\right)}{x} d x\)
Solution:
\(\int \frac{(x+2)\left(4 x^2-5\right)}{x} d x\)
\(=\int \frac{4 x^3+8 x^2-5 x-10}{x} d x\)
\(=4 \int 4 x^2 d x+8 \int x d x-\int 5 d x-\int \frac{10}{x} d x\)
\(=4 \frac{x^3}{3}+8 \frac{x^2}{2}-5 x-10 \log |x|+C\)
\(=4 \frac{x^3}{3}+4 x^2-5 x-10 \log |x|+C\)

Question 17.
\(\int \frac{x^2}{4+x^2} d x\)
Solution:
\(\text { Let } I=\int \frac{x^2}{4+x^2} d x\)
\(=\int \frac{4+x^2-4}{4+x^2} d x=\int\left[1-\frac{4}{x^2+4}\right] d x\)
\(=x-4 \int \frac{d x}{x^2+2^2}\)
\(=x-\frac{4}{2} \tan ^{-1} \frac{x}{2}+C\)
\(=x-2 \tan ^{-1} \frac{x}{2}+C\)

Question 18.
\(\int \frac{x^4}{1+x^2} d x\)
Solution:
Let I = \(\int \frac{x^4}{1+x^2} d x\) = \(\int \frac{x^4-1+1}{1+x^2} d x\)
\(=\int \frac{x^4-1}{1+x^2} d x+\int \frac{d x}{1+x^2}\)
\(=\int \frac{\left(x^2-1\right)\left(x^2+1\right)}{1+x^2} d x+\int \frac{d x}{1+x^2}\)
\(=\int \frac{\left(x^2-1\right)\left(x^2+1\right)}{1+x^2} d x+\int \frac{d x}{1+x^2}\)
\(=\frac{x^3}{x}-x+\tan ^{-1} x+c\)

Question 19.
\(\int \frac{x^6-1}{x^2+1} d x\)
Solution:
\(\text { Let } \mathrm{I}=\int \frac{x^6-1}{x^2+1} d x\)
\(=\int \frac{\left(x^6+1-2\right)}{x^2+1} d x\)
\(=\int \frac{\left(x^2\right)^2+1}{x^2+1} d x-\int \frac{2 d x}{x^2+1}\)
\(=\int\left[\left(x^4-x^2+1\right)+\frac{-2}{x^2+1}\right] d x\)
\(=\frac{x^5}{5}-\frac{x^3}{3}+x-2 \tan ^{-1} x+c\)

Question 20.
\(\int \frac{e^{6 \log x}-e^{5 \log x}}{e^{5 \log x}-e^{3 \log x}} d x\)
Solution:
Let I = \(\int \frac{e^{6 \log x}-e^{5 \log x}}{e^{5 \log x}-e^{3 \log x}} d x\)
\(=\int \frac{e^{\log x^6}-e^{\log x^5}}{e^{\log x^5}-e^{\log x^3}} d x\) = \(\int \frac{x^6-x^5}{x^5-x^3} d x\)
\(=\int \frac{x^5(x-1)}{x^3\left(x^2-1\right)} d x=\int \frac{x^2}{(x+1)} d x\)
\(=\int \frac{x^2-1+1}{x+1} d x=\int\left[(x-1)+\frac{1}{x+1}\right] d x\)
\(=\frac{x^2}{2}-x+\log (x+1)+c .\)