int frac{d x}{xleft(x^4+1right)}= | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) To solve the integral $I = \int \frac{dx}{x(x^4+1)}$,multiply the numerator and denominator by $x^3$: $I = \int \frac{x^3 dx}{x^4(x^4+1)}$ Let $x^

Vedclass · Accepted Answer

$\frac{1}{4} \log \left(\frac{x^4}{x^4+1}\right)+c$

Vedclass · Answer

(D) To solve the integral $I = \int \frac{dx}{x(x^4+1)}$,multiply the numerator and denominator by $x^3$: $I = \int \frac{x^3 dx}{x^4(x^4+1)}$ Let $x^4 = t$,then $4x^3 dx = dt$,which implies $x^3 dx = \frac{dt}{4}$. Substituting these into the integral: $I = \frac{1}{4} \int \frac{dt}{t(t+1)}$ Using partial fractions,$\frac{1}{t(t+1)} = \frac{1}{t} - \frac{1}{t+1}$. $I = \frac{1}{4} \int \left( \frac{1}{t} - \frac{1}{t+1} \right) dt$ $I = \frac{1}{4} [\log|t| - \log|t+1|] + C$ $I = \frac{1}{4} \log \left| \frac{t}{t+1} \right| + C$ Substituting $t = x^4$ back: $I = \frac{1}{4} \log \left( \frac{x^4}{x^4+1} \right) + C$

$\int \frac{d x}{x\left(x^4+1\right)}=$

Similar Questions

$\int \frac{(\sin^{-1} x)^{\frac{3}{2}}}{\sqrt{1-x^2}} dx =$

$\int \frac{\csc^2 x}{1 + \cot x} dx = $

If $\int \frac{\sin \left(x-\frac{\pi}{4}\right)}{2+\sin 2 x} d x=-\frac{1}{\sqrt{2}} \tan ^{-1}(f(x))+C$,then $f(x)=$

If $n \geq 2$ is a natural number and $0 < \theta < \frac{\pi}{2}$,then $\int \frac{(\cos ^n \theta-\cos \theta)^{1 / n}}{\cos ^{n+1} \theta} \sin \theta d \theta =$

$\int \cos ^3 x \cdot e^{\log (\sin x)} d x=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module