Question
Evaluate : $\int \frac{1}{a^2-b^2 x^2} \cdot d x$
✓
Answer
\begin{aligned}
& \mathrm{I}=\int \frac{1}{b^2\left(\frac{a^2}{b^2}-x^2\right)} \cdot d x \\
& =\frac{1}{b^2} \cdot \int \frac{1}{\left(\frac{a}{b}\right)^2-x^2} \cdot d x \\
& \because \quad \int \frac{1}{a^2-x^2} \cdot d x=\frac{1}{2 a} \log \left(\frac{a+x}{a-x}\right)+c \\
& \mathrm{I}=\frac{1}{b^2} \cdot \frac{1}{2\left(\frac{a}{b}\right)} \cdot \log \left(\frac{\frac{a}{b}+x}{\frac{a}{b}-x}\right)+c \\
& =\frac{1}{b^2} \cdot \frac{1}{2\left(\frac{a}{b}\right)} \cdot \log \left(\frac{\frac{a}{b}+x}{\frac{a}{b}-x}\right)+c \\
& =\frac{1}{2 a b} \cdot \log \left(\frac{a+b x}{a-b x}\right)+c \\
\end{aligned}
& \mathrm{I}=\int \frac{1}{b^2\left(\frac{a^2}{b^2}-x^2\right)} \cdot d x \\
& =\frac{1}{b^2} \cdot \int \frac{1}{\left(\frac{a}{b}\right)^2-x^2} \cdot d x \\
& \because \quad \int \frac{1}{a^2-x^2} \cdot d x=\frac{1}{2 a} \log \left(\frac{a+x}{a-x}\right)+c \\
& \mathrm{I}=\frac{1}{b^2} \cdot \frac{1}{2\left(\frac{a}{b}\right)} \cdot \log \left(\frac{\frac{a}{b}+x}{\frac{a}{b}-x}\right)+c \\
& =\frac{1}{b^2} \cdot \frac{1}{2\left(\frac{a}{b}\right)} \cdot \log \left(\frac{\frac{a}{b}+x}{\frac{a}{b}-x}\right)+c \\
& =\frac{1}{2 a b} \cdot \log \left(\frac{a+b x}{a-b x}\right)+c \\
\end{aligned}
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The principal solution of the equation $\cot x=-\sqrt{3}$ is
→Find $\bar{a} \cdot(\bar{b} \times \bar{c})$, if $\bar{a}=3 \hat{i}-\hat{j}+4 \hat{k} \bar{b}=2 \hat{i}+3 \hat{j}-\hat{k}$ and $\bar{c}=-5 \hat{i}+2 \hat{j}+3 \hat{k}$
→Write converse and inverse of the following statement: "If $x>y$ then $x^2>y^{2 }$"
→Evaluate :
→$\int_0^{\pi / 2} \log \tan x \cdot d x$
If $\begin{bmatrix}\text{x}&2\end{bmatrix}\begin{bmatrix}3\\4\end{bmatrix}=2,$ find x.
→Let X be the amount of time for which a book is taken out of the library by a randomly selected students and suppose X has p.d.f.
f(x) = 0.5x, for 0 ≤ x ≤ 2 and = 0 otherwise.
Calculate:
P(0.5 ≤ x ≤ 1.5)
f(x) = 0.5x, for 0 ≤ x ≤ 2 and = 0 otherwise.
Calculate:
P(0.5 ≤ x ≤ 1.5)
Check whether the following switching circuits are logically equivalent – Justify.
Write the value of $\cos^{-1}(\cos6).$
→What is the cosine of the angle with the vector $\sqrt2\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ makes with y-axis?
→Find the points on the curve given by $y=x^3-6 x^2+x+3$ where the tangents are parallel to the line $y=x+5$.
→