Question

Evaluate the following integrals: (i) \(\int \frac{d x}{x^{2}\left(1+x^{5}\right)^{4 / 5}}\) (ii) \(\int \frac{x^{2}-1}{x \sqrt{\left(1+x^{4}\right)}} \mathrm{dx}\)

Short answer

In summary: (i) The integral of \(\int \frac{dx}{x^{2}\left(1+x^{5}\right)^{4 / 5}}\) simplifies to become \(-\frac{1}{25}(1+x^5)^{1/5}+C\). (ii) The integral of \(\int \frac{x^2-1}{x\sqrt{1+x^4}} dx\) simplifies to become \(\sqrt{1+x^4}+C\).

Step-by-step solution

Problem(i): Substitution

Let's use a substitution technique to deal with the given integral. Let \(u=x^5\), then the differential \(du\) is given by \(du = 5x^4dx\). We will express the integral in terms of \(u\) and perform the integration. Now, let's replace the variables in the integral: \(\int \frac{dx}{x^{2}\left(1+x^{5}\right)^{4 / 5}}=\int \frac{1}{x^{2}\left(1+u\right)^{4 / 5}} \frac{1}{5x^4}du\).

Problem(i): Simplification

Next, we will simplify the expression and perform the integration: \(\int \frac{1}{x^2\left(1+u\right)^{4 / 5}} \frac{1}{5x^4}du = \frac{1}{5}\int \frac{1}{u^{2/5}(1+u)^{4/5}} du\).

Problem(i): Integration

Now, we can integrate the simplified integral: \(\frac{1}{5}\int \frac{1}{u^{2/5}(1+u)^{4/5}} du=-\frac{1}{5}\int \frac{(1+u)^{1/5}}{u^{2/5}} (1+u)^{-1} du\). Let \(v=(1+u)^{1/5}\), so \(dv=\frac{1}{5}(1+u)^{-4/5}du\). Thus, the integral becomes \(-\int v^{4} dv=-\frac{1}{5}\int v^4 dv\).

Problem(i): Final Result

Now, we integrate and revert to the original variable: \(-\frac{1}{5}\int v^4 dv=-\frac{1}{25}v^5+C=-\frac{1}{25}(1+u)^{1/5}+C=-\frac{1}{25}(1+x^5)^{1/5}+C\).

Problem(ii): Simplification

First, we will simplify the integrand by breaking it into two terms: \(\int \frac{x^2-1}{x\sqrt{1+x^4}} dx = \int \frac{x^2}{x\sqrt{1+x^4}} dx - \int \frac{1}{x\sqrt{1+x^4}} dx\).

Problem(ii): Further Simplification

Now, we simplify each integral: \(\int \frac{x^2}{x\sqrt{1+x^4}} dx - \int \frac{1}{x\sqrt{1+x^4}}dx = \int \frac{x}{\sqrt{1+x^4}} dx - \int \frac{1}{x\sqrt{1+x^4}} dx\).

Problem(ii): Substitution

Let's use another substitution technique with \(w = 1+x^4\), then the differential \(dw\) is given by \(dw = 4x^3dx\). We will express the integrals in terms of \(w\) and perform the integration. Now, let's replace the variables in the integrals: \(\int \frac{x}{\sqrt{1+x^4}}dx - \int \frac{1}{x\sqrt{1+x^4}}dx = \int \frac{1}{x^2\sqrt{w}}\frac{1}{4x^3}dw - \int \frac{1}{\sqrt{w}x^3} \frac{1}{4x^3}dw\).

Problem(ii): Integration

Now, we can perform the integration: \(\int \frac{1}{x^2\sqrt{w}}\frac{1}{4x^3}dw - \int \frac{1}{\sqrt{w}x^3} \frac{1}{4x^3}dw = \frac{1}{4}\int \frac{dw}{w\sqrt{w}} - \frac{1}{4}\int \frac{dw}{w\sqrt{w}}\).

Problem(ii): Final Result

Finally, we integrate and revert to the original variable: \(\frac{1}{4}\int \frac{dw}{w\sqrt{w}} - \frac{1}{4}\int \frac{dw}{w\sqrt{w}} = \frac{1}{2}\int \frac{dw}{w\sqrt{w}} + C = \frac{1}{2}\int \frac{dw}{w^{3/2}} + C\). Now, integrating and reverting back to \(x\) variable: \(\frac{1}{2}\int \frac{dw}{w^{3/2}} +C = \frac{1}{2}\left[2\sqrt{w}\right]+C= \sqrt{w}+C=\sqrt{1+x^4}+C\).