Exercise 11.3-Additional Questions - Chapter 11 Integral Calculus 11th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Additional Problems
Question 1 .
$
5 x^4+3(2 x+3)^4-6(4-3 x)^5
$

Solution:
$
\begin{aligned}
\int\left[5 x^4+3(2 x+3)^4\right. & \left.-6(4-3 x)^5\right] d x \\
& =5 \int x^4 d x+3 \int(2 x+3)^4 d x-6 \int(4-3 x)^5 d x \\
& =5 \cdot \frac{x^5}{5}+3 \cdot \frac{1}{2} \frac{(2 x+3)^5}{5}-6 \cdot \frac{1}{(-3)} \frac{(4-3 x)^6}{6}+c \\
& =x^5+\frac{3}{10}(2 x+3)^5+\frac{1}{3}(4-3 x)^6+c
\end{aligned}
$
Question 2.
$
4-\frac{5}{x+2}+3 \cos 2 x
$
Solution:
$
\begin{aligned}
& \int\left[4-\frac{5}{x+2}+3 \cos 2 x\right] d x \\
&=4 \int d x-5 \int \frac{1}{x+2} d x+3 \int \cos 2 x d x \\
&=4 x-5 \log (x+2)+\frac{3}{2} \sin 2 x+c .
\end{aligned}
$
Question 3.
$
\begin{aligned}
& 9 \operatorname{cosec}^2(\mathrm{px}-q)-6(1-\mathrm{x})^4+4 \mathrm{e}^{3-4 \mathrm{x}} \\
& \text { Solution: } \begin{aligned}
\int\left[p \operatorname{cosec}^2(p x-q)\right. & \left.-6(1-x)^4+4 e^{3-4 x}\right] d x \\
& =p \int \operatorname{cosec}^2(p x-q) d x-6 \int(1-x)^4 d x+4 \int e^{3-4 x} d x \\
& =p \times \frac{1}{p}-\cot (p x-q)-\frac{6 \cdot(1-x)^5}{(-1)(5)}+4 \cdot \frac{e^{3-4 x}}{(-4)}+c \\
& =-\cot (p x-q)+\frac{6}{5}(1-x)^5-e^{3-4 x}+c .
\end{aligned}
\end{aligned}
$

Question 4.
$
\frac{4}{(3+4 x)}+(10 \mathrm{x}+3)^9-3 \operatorname{cosec}(2 \mathrm{x}+3) \cot (2 \mathrm{x}+3)
$
Solution:

$
\begin{aligned}
\int\left[\frac{4}{(3+4 x)}+(10 x\right. & \left.+3)^9-3 \operatorname{cosec}(2 x+3) \cot (2 x+3)\right] d x \\
& =4 \int \frac{1}{3+4 x} d x+\int(10 x+3)^9 d x-3 \int \operatorname{cosec}(2 x+3) \cot (2 x+3) d x \\
& =4 \times \frac{1}{4} \log (3+4 x)+\frac{1}{10} \frac{(10 x+3)^{10}}{10}-3 \times \frac{1}{2}(-\operatorname{cosec}(2 x+3))+c \\
= & \log (3+4 x)+\frac{1}{100}(10 x+3)^{10}+\frac{3}{2} \operatorname{cosec}(2 x+3)+c
\end{aligned}
$
Question 5.
$\mathrm{a} \sec ^2(\mathrm{bx}+\mathrm{c})+\frac{q}{e^{-m x}}$
Solution:
$
\begin{aligned}
\int\left[a \sec ^2(b x+c)+\right. & \left.\frac{q}{e^{l-m x}}\right] d x \\
& =a \int \sec ^2(b x+c) d x+q \int e^{m x-1} d x \\
& =a \cdot \frac{1}{b} \tan (b x+c)+q \cdot \frac{1}{m} e^{m x-l} d x \\
& =\frac{a}{b} \tan (b x+c)+\frac{q}{m e^{l-m x}}+c
\end{aligned}
$
Question 6.
$
\frac{1}{\left(3+\frac{2}{3} x\right)}-\frac{2}{3} \cos \left(x-\frac{2}{3}\right)+3\left(\frac{x}{3}+4\right)^6
$

Solution:
$
\begin{aligned}
& \int\left[\frac{1}{\left(3+\frac{2}{3} x\right)}-\frac{2}{3} \cos \left(x-\frac{2}{3}\right)+3\left(\frac{x}{3}+4\right)^6\right] d x \\
& =\int \frac{1}{\left(3+\frac{2}{3} x\right)} d x-\frac{2}{3} \int \cos \left(x-\frac{2}{3}\right) d x+3 \int\left(\frac{x}{3}+4\right)^6 d x \\
& =\frac{1}{2} \log \left(3+\frac{2}{3} x\right)-\frac{2}{3} \sin \left(x-\frac{2}{3}\right)+3 \cdot \frac{1}{1 / 3} \frac{(x / 3+4)^7}{7}+c \\
& =\frac{1}{2} \log \left(3+\frac{2}{3} x\right)-\frac{2}{3} \sin \left(x-\frac{2}{3}\right)+\frac{9}{7}\left(\frac{x}{3}+4\right)^7+c \text {. } \\
&
\end{aligned}
$
Question 7.
$
\begin{aligned}
& 7 \sin \frac{x}{7}-8 \sec ^2\left(4-\frac{x}{4}\right)+10\left(\frac{2 x}{5}-4\right)^{\frac{3}{2}} \\
& \text { Solution: } \\
& \int\left[7 \sin \frac{x}{7}-8 \sec ^2\left(4-\frac{x}{4}\right)+10\left(\frac{2 x}{5}-4\right)^{\frac{3}{2}}\right] d x \\
& =7 \int \sin \frac{x}{7} d x-8 \int \sec ^2\left(4-\frac{x}{4}\right) d x+10 \int\left(\frac{2 x}{5}-4\right)^{\frac{3}{2}} d x \\
& =7 \cdot \frac{1}{1 / 7}\left(-\cos \frac{x}{7}\right)-8 \frac{1}{(-1 / 4)} \tan \left(4-\frac{x}{4}\right)+10 \cdot \frac{1}{\frac{2}{5}} \frac{\left(\frac{2 x}{5}-4\right)^{\frac{3}{2}+1}}{\frac{3}{2}+1}+c \\
& =-49 \cos \frac{x}{7}+32 \tan \left(4-\frac{x}{4}\right)+\frac{10 \times 5}{2 \times \frac{5}{2}}\left(\frac{2 x}{5}-4\right)^{\frac{5}{2}}+c \\
&
\end{aligned}
$

$=-49 \cos \frac{x}{7}+32 \tan \left(4-\frac{x}{4}\right)+10\left(\frac{2 x}{5}-4\right)^{\frac{5}{2}}+c$