Q.Integrate the function $\frac{1}{x^2(x^4+1)^{3/4}}$

Concept: Partial Fraction Decomposition – but here we use a clever substitution to avoid it.

We have

Step 1: Rewrite the integrand by factoring $x^4$ inside the power:

So the integral becomes

Step 2: Substitute $t = 1 + \frac{1}{x^4}$. Then $dt = -\frac{4}{x^5} dx$, so $\frac{dx}{x^5} = -\frac{dt}{4}$.

Step 3: The integral becomes …

We integrate $\displaystyle\int\frac{dx}{x^2\,(x^4+1)^{3/4}}$.

1. Pull $x^4$ out of the bracket. Since $(x^4+1)^{3/4}=x^3\!\left(1+x^{-4}\right)^{3/4}$,

2. Substitute $t=1+x^{-4}$, so $dt=-4x^{-5}\,dx\Rightarrow \dfrac{dx}{x^5}=-\dfrac14\,dt$:

$\int t^{-3/4}\left(-\frac14\right)dt=-\frac14\int t^{-3/4}\,dt.$ …

Method: Factor the dominant power out of the bracket, then substitute

Use this for $\displaystyle\int x^{m}\,(ax^{n}+b)^{p}\,dx$ with a fractional $p$: pulling $x^{n}$ out of the bracket exposes a substitution $t=b+ax^{-n}$ (or $t=a+bx^{-n}$).

Steps

Step 1: Pull the highest power of $x$ out of the bracket.

Write $(x^{n}+1)^{p}=x^{np}\bigl(1+x^{-n}\bigr)^{p}$ so the bracket depends only on $x^{-n}$.

Step 2: Choose $t=1+x^{-n}$. …

Common Mistakes

Mistake 1: Not factoring $x^4$ out of $(x^4+1)^{3/4}$.

Why it's wrong: without rewriting $(x^4+1)^{3/4}=x^3(1+x^{-4})^{3/4}$, no clean substitution appears and the integral looks intractable. Correct approach: pull the dominant power out of the bracket first.

Mistake 2: Losing the constant $-\tfrac14$ from $dt$.

Why it's wrong: with $t=1+x^{-4}$, $dt=-4x^{-5}\,dx$, so $\tfrac{dx}{x^5}=-\tfrac14\,dt$; missing the $-\tfrac14$ scales the answer wrongly. Correct approach: carry the factor through. …