Evaluate the integral. int x sqrt(5x-1) dx

Evaluate the integral. int x sqrt(5x-1) dx

Question
Integrals
asked 2020-10-20
Evaluate the integral. \(\displaystyle\int{x}\sqrt{{{5}{x}-{1}}}{\left.{d}{x}\right.}\)

Answers (1)

2020-10-21
The key here is to do a change of variables. Specifically, any time you see a square root, you either want to convert it to something squared under the square root or just set u= whatever is under it. In this case, we'll do the latter.
Let u=5x−1.Then, because we know \(\displaystyle{d}{u}=\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}{\left.{d}{x}\right.}\), we see that du=5dx which is equivalent to \(\displaystyle{k}{\left.{d}{x}\right.}=\frac{{1}}{{d}}{u}\). Substituting these into the integral, we get.
\(\displaystyle\int{x}\sqrt{{u}}\cdot\frac{{1}}{{5}}{d}{u}\)
This is part of what we want, but it still has that xx there and we want to totally convert to uu's. Well, remember that we set u=5x−1. Let's solve that for x in terms of u:
\(\displaystyle{5}{x}={u}+{1}\Rightarrow{x}=\frac{{{u}+{1}}}{{5}}\)
Substituting this in, we get
\(\displaystyle\int\frac{{{u}+{1}}}{{5}}\sqrt{{u}}\cdot\frac{{1}}{{5}}{d}{u}=\frac{{1}}{{25}}\int{\left({u}+{1}\right)}\sqrt{{u}}{d}{u}=\frac{{1}}{{25}}\int{\left({u}^{{\frac{{3}}{{2}}}}+{u}^{{\frac{{1}}{{2}}}}\right)}{d}{u}\)
0

Relevant Questions

asked 2020-11-22
Evaluate the integral \(\int \frac{1}{1+\frac{x}{2}^2}dx\)
asked 2021-02-02
Evaluate the integral \(\displaystyle\int{\frac{{{1}}}{{{1}+{\frac{{{x}}}{{{2}}}}^{{2}}}}}{\left.{d}{x}\right.}\)
asked 2021-02-21
Evaluate the integral:
\(\displaystyle\int\int\int_{{E}}{\left({x}{y}+{z}^{{2}}\right)}{d}{V}\),
where \(\displaystyle{E}={\left\lbrace{\left({x},{y},{z}\right)}{\mid}{0}\le{x}\le{2},{0}\le{y}\le{1},{0}\le{z}{<}{3}\right\rbrace}\)
asked 2021-02-06
Evaluate the triple integral \(\displaystyle\int\int\int_{{E}}{3}{y}{d}{V}\),where
\(\displaystyle{E}={\left\lbrace{\left({x},{y},{z}\right)}{\mid}{0}\le{x}\le{2},{0}\le{y}\le\sqrt{{{4}-{x}^{{2}}}},{0}\le{z}\le{x}\right\rbrace}\)
asked 2021-02-22
Evaluate the following derivatives.
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{7}}}^{{{x}}}}\sqrt{{{1}+{t}^{{{4}}}+{t}^{{{6}}}}}{\left.{d}{t}\right.}\)
asked 2021-01-13
Evaluate the integral ∫(2/3sqrtx)dx
asked 2020-11-03
Evaluate the surface integral
\(\displaystyle\int_{{S}}{F}\cdot{d}{S}\)
for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation.
F(x,y,z)=xi+2yj+3zk
S is the cube with vertices \(\displaystyle{\left(\pm{1},\pm{1},\pm{1}\right)}\)
asked 2020-11-11
Give the correct answer and solve the given equation Evaluate \(\displaystyle\int{x}^{3}{\left({\sqrt[{3}]{{{1}-{x}^{2}}}}\right)}{\left.{d}{x}\right.}\)
asked 2021-02-09
Evaluate the integral \(\displaystyle\int\frac{x}{{2}}{\left.{d}{x}\right.}\)
asked 2021-03-09
Evaluate the integral \(\displaystyle\int{e}^{{{3}{x}}} \cos{{2}}{x}{\left.{d}{x}\right.}\)
...