Evaluate the follwing intregals: 1 x^4-1 dx - Vidyadip

Question

Evaluate the follwing intregals:
$\int\frac{1}{\text{x}^4-1}\text{ dx}$

✓

Answer

The Evaluate the integral follow the steps,
$\int\frac{1}{(\text{x}^4-1)}\text{ dx}$
Let $\frac{1}{(\text{x}^4-1)}=\frac{\text{A}}{\text{x}-1}+\frac{\text{B}}{\text{x}-1}+\frac{\text{C}}{\text{x}^2+1}$
$1={\text{A}}{(\text{x}-1})(\text{x}^2+1)+\text{B}(\text{x}+1)(\text{x}^2+1)+\text{C}(\text{x}+1)(\text{x}-1)$
$\text{For x}=1,\text{ B}=\frac{1}{4}$
$\text{For x}=-1,\text{ A}=-\frac{1}{4}$
$\text{For x}=0,\text{ C}=-\frac{1}{2}$
Therefore,
$\int\frac{1}{(\text{x}^4-1)}\text{ dx}=-\frac{1}{4}\int\frac{\text{dx}}{\text{x}+1}+\frac{1}{4}\frac{\text{dx}}{\text{x}-1}-\frac{1}{2}\int\frac{\text{dx}}{\text{x}^2+1}$
$=-\frac{1}{4}\ln\big|(\text{x}+1)\big|+\frac{1}{4}\ln\big|(\text{x}-1)\big|-\frac{1}{2}\tan^{-1}\text{x}+\text{C}$
$=\frac{1}{4}\ln\Big|\frac{\text{x}-1}{\text{x}+1}\Big|-\frac{1}{2}\tan^{-1}\text{x}+\text{C}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Let $\text{f}\text{(x)}=\frac{\log\Big(1+\frac{\text{x}}{\text{a}}\Big)-\log\Big(1-\frac{\text{x}}{\text{b}}\Big)}{\text{x}},\text{x}\neq0$ Find the value of f at x = 0. So that f becomes continuous at x = 0.

Evaluate the following integrals:
$\int\frac{1}{4\text{x}^2+12\text{x}+5}\text{dx}$

Prove that $2\tan^{-1}\bigg(\sqrt{\frac{\text{a}-\text{b}}{\text{a}+\text{b}}}\tan\frac{\theta}{2}\bigg)=\cos^{-1}\Big(\frac{\text{a}\cos\theta+b}{\text{a}+\text{b}\cos\theta}\Big)$

Differentiate $\cos^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)$ with respect to $\cos^{-1}\Big(\frac{1-\text{x}^2}{1+\text{x}^2}\Big),$ if $0<\text{x}<1$

Find the area bounded by the lines y = 4x + 5, y 5 - x and 4y = x + 5.

Find the equation of the plane through the point $2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}$ and passing throught the line of intersection of the planes $\vec{\text{r}}\cdot(\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}})=0$ and $\vec{\text{r}}\cdot(\hat{\text{j}}+2\hat{\text{k}})=0.$

Evaluate the following integrals as limit of sum:$\int\limits^{3}_{1}\big(2\text{x}^2+5\text{x}\big)\text{dx}$

If $\sin^{-1}\text{x}+\sin^{-1}\text{y}+\sin^{-1}\text{z}+\sin^{-1}\text{t}=2\pi,$ then find the value of $x^2 + y^2 + z^2 + t^2$

If $\sqrt{\text{y}+\text{x}}+\sqrt{\text{y}-\text{x}}=\text{c},$ show that $\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}-\sqrt{\frac{\text{y}^2}{\text{x}^2}-1}$

Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}|\text{x}|+3,&\text{if }\text{ x}\geq-3\\-2\text{x},&\text{if }-3<\text{ x}<3\\6\text{x}+2,&\text{if }\text{ x}>3\end{cases}$