How does the these steps came?

Question

I'm learning single variable calculus right now and at current about integration. I'm confused in a problem from sometime especially in two steps which I've circle. Please help. thankyou in advance.

Answer 1

Just to have an answer to this question. The two steps you mark are indeed wrong. As pointed out by @MrYouMath, $$ (x^4+1)^{3/4}=(x^4(1+1/x^4))^{3/4}=x^3(1+1/x^4)^{3/4}. $$ So there is a factor of $x^3$ missing.

In the second step, the $x^3$ should be $x^5$.

Answer 2

The first step is wrong: $$\int\frac{dx}{x^2(x^4+1)^{3/4}}=\int{\frac{dx}{x^2(x^4(1+\frac{1}{x^4}))^{3/4}}}=\int{\frac{dx}{x^2\cdot (x^4)^{3/4}(1+\frac{1}{x^4})^{3/4}}}=\int{\frac{dx}{x^5(1+\frac{1}{x^4})^{3/4}}}$$