# Evaluate the following integral. \int \frac{x}{\sqrt{x-4}}dx

Question
Applications of integrals
Evaluate the following integral.
$$\displaystyle\int{\frac{{{x}}}{{\sqrt{{{x}-{4}}}}}}{\left.{d}{x}\right.}$$

2021-02-15
Step 1
Consider the integrals,
$$\displaystyle\int{\frac{{{x}}}{{\sqrt{{{x}-{4}}}}}}{\left.{d}{x}\right.}$$
Suppose that,
$$\displaystyle\sqrt{{{x}-{4}}}={t}$$
Differentiating with respect to "x"
$$\displaystyle{\frac{{{1}}}{{{2}\sqrt{{{x}-{4}}}}}}{\left.{d}{x}\right.}={\left.{d}{t}\right.}$$
$$\displaystyle{\frac{{{1}}}{{\sqrt{{{x}-{4}}}}}}{\left.{d}{x}\right.}={2}{\left.{d}{t}\right.}$$
Step 2
Substitute all value in given integrals,
$$\displaystyle\int{\frac{{{x}}}{{\sqrt{{{x}-{4}}}}}}{\left.{d}{x}\right.}=\int{\left({t}^{{{2}}}+{4}\right)}{2}\ {\left.{d}{t}\right.}$$
$$\displaystyle={2}\int{\left({t}^{{{2}}}+{4}\right)}{\left.{d}{t}\right.}$$
$$\displaystyle={2}{\left[{\frac{{{t}^{{{3}}}}}{{{3}}}}+{4}{t}\right]}+{C}$$
$$\displaystyle={\frac{{{2}{t}^{{{3}}}}}{{{3}}}}+{8}{t}+{C}$$
$$\displaystyle={\frac{{{2}}}{{{3}}}}{\left({x}-{4}\right)}^{{{\frac{{{3}}}{{{2}}}}}}+{8}\sqrt{{{\left({x}-{4}\right)}}}+{C}$$

### Relevant Questions

Evaluate the following integral.
$$\int \frac{3x^{2}+\sqrt{x}}{\sqrt{x}}dx$$
Evaluate the following integral: $$\int \frac{x+3}{x-1}dx$$
Use a change of variables to evaluate the following integral.
$$\int-(\cos^{7}x-5\cos^{5}x-\cos x)\sin x dx$$
Evaluate each of the following integrals.
$$\int\frac{e^{x}}{1+e^{x}}dx$$
Evaluate the following integral: $$\int \frac{(y-3)}{y^{2}-6y+1}$$
Evaluate the following integral: $$\int\frac{vdv}{6v^{2}-1}$$
Find the indefinite integral $$\int \ln(\frac{x}{3})dx$$ (a) using a table of integrals and (b) using the Integration by parts method.
$$\int_{0}^{1}t^{\frac{5}{2}}(\sqrt{t}-3t)dt$$
$$\int_{1}^{2}\frac{(x+1)^{2}}{x}dx$$
$$\int 2x^{3}+3x-2dx$$