Question

Evaluate the following integrals : $$ \int x^{34}\left(1+x^{78}\right)^{1 / 2} d x $$

Short answer

Question: Evaluate the integral ∫x^34(1+x^78)^(1/2) dx Answer: The evaluated integral is ∫x^34(1+x^78)^(1/2) dx = (1+x^78)/(117) (1 + x^78)^(1/2) + C, where C is the constant of integration.

Step-by-step solution

Choose a substitution function

Let's make a substitution: $$ u = (1+x^{78})^{1/2} $$

Differentiate the substitution function

Now, differentiate \(u\) with respect to \(x\). We get: $$ \frac{d u}{d x} = \frac{1}{2}(1 + x^{78})^{-\frac{1}{2}} \cdot 78 x^{77} $$

Rearrange to find dx

We want to rearrange this equation to solve for \(dx\). In order to do that, we first need to express \(x^{77}\) in terms of \(u\). If we square both sides of our substitution, we get: $$ u^2 = 1 + x^{78} $$ Subtract 1 from both sides: $$ u^2 - 1 = x^{78} $$ Take the 77th root of both sides: $$ x^{77} = \sqrt[77]{u^2 - 1} $$ Now we can substitute this into our equation for \(du/dx\) and solve for \(dx\): $$ \frac{d u}{d x} = \frac{1}{2}(1 + x^{78})^{-\frac{1}{2}} \cdot 78 \sqrt[77]{u^2 - 1} $$ $$ \frac{d u}{d x} = \frac{1}{2} u^{-1} \cdot 78 \sqrt[77]{u^2 - 1} $$ Rearranging to solve for \(dx\): $$ dx = \frac{2}{78} \cdot u \cdot \frac{du}{\sqrt[77]{u^2 - 1}} $$

Substitute back into original integral

Now that we have our substitution and the expression for \(dx\), we can rewrite the original integral in terms of \(u\): $$ \int x^{34}\left(1+x^{78}\right)^{1 / 2} dx = \int \left(\sqrt[77]{u^2 - 1}\right)^{34} u \cdot \frac{2}{78} \cdot u \cdot \frac{d u}{\sqrt[77]{u^2 - 1}} $$

Simplify the integral

Now we can simplify the integral: $$ \frac{2}{78} \int u^2 du = \frac{u^3}{117} $$

Substitute back \(x\) for \(u\)

Now we will substitute back \(u = (1 + x^{78})^{1/2}\) to find our final answer: $$ \frac{u^3}{117} = \frac{\left((1 + x^{78})^{1/2}\right)^3}{117} = \frac{1+x^{78}}{117} (1 + x^{78})^{1 / 2} $$ So, the evaluated integral is: $$ \int x^{34}\left(1+x^{78}\right)^{1 / 2} dx = \frac{1+x^{78}}{117} (1 + x^{78})^{1 / 2} + C $$ Where, \(C\) is the constant of integration.