Indefinite IntegrationeasyFree

Integral of (x³−1)/((x⁴+1)(x+1)) | JEE

JEE Maths question with a full step-by-step solution.

Question

\displaystyle\int \dfrac{x^{3}-1}{(x^{4}+1)(x+1)}\,dx

\dfrac{1}{4}\ln(1+x^{4})+\dfrac{1}{3}\ln(1+x^{3})+C

\dfrac{1}{4}\ln(1+x^{4})-\dfrac{1}{3}\ln(1+x^{3})+C

\dfrac{1}{4}\ln(1+x^{4})-\ln(1+x)+C

correct

\dfrac{1}{4}\ln(1+x^{4})+\ln(1+x)+C

Solution

Step 1: Split the numerator so the fraction breaks into standard pieces. Add and subtract

x^{4}

x^{3}-1=(x^{4}+x^{3})-(x^{4}+1)=x^{3}(x+1)-(x^{4}+1)

Step 2: Divide each part by the denominator

(x^{4}+1)(x+1)

\dfrac{x^{3}-1}{(x^{4}+1)(x+1)}=\dfrac{x^{3}(x+1)}{(x^{4}+1)(x+1)}-\dfrac{x^{4}+1}{(x^{4}+1)(x+1)}=\dfrac{x^{3}}{x^{4}+1}-\dfrac{1}{x+1}

Step 3: Integrate the first piece. Since

x^{3}=\dfrac{1}{4}\dfrac{d}{dx}(x^{4}+1)

\int\dfrac{x^{3}}{x^{4}+1}\,dx=\dfrac{1}{4}\ln(x^{4}+1)

Step 4: Integrate the second piece and combine:

\int\dfrac{1}{x+1}\,dx=\ln(x+1)\;\Rightarrow\; I=\dfrac{1}{4}\ln(x^{4}+1)-\ln(x+1)+C

Correct answer: (3)

Still stuck on this question?Ask your doubt on WhatsApp

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.