Evaluate the integral. int sqrt x(x^3+x/2)dx

Evaluate the integral. int sqrt x(x^3+x/2)dx

Question
Applications of integrals
asked 2020-10-21
Evaluate the integral.
\(\displaystyle\int\sqrt{{x}}{\left({x}^{{3}}+\frac{{x}}{{2}}\right)}{\left.{d}{x}\right.}\)

Answers (1)

2020-10-22
Given That
\(\displaystyle\int\sqrt{{x}}{\left({x}^{{3}}+\frac{{x}}{{2}}\right)}{\left.{d}{x}\right.}\)
\(\displaystyle=\int{\left({x}^{{{3}+\frac{{1}}{{2}}}}+\frac{{{x}^{{\frac{{1}}{{2}}}}\cdot{x}}}{{2}}\right)}{\left.{d}{x}\right.}\)
\(\displaystyle=\int{\left({x}^{{\frac{{7}}{{2}}}}\right)}+\frac{{{x}^{{\frac{{3}}{{2}}}}}}{{2}}{)}{\left.{d}{x}\right.}\)
\(\displaystyle=\frac{{{x}^{{\frac{{7}}{{2}}}}+{1}}}{{\frac{{7}}{{2}}+{1}}}+\frac{{{x}^{{\frac{{3}}{{2}}}}+{1}}}{{\frac{{3}}{{2}}+{1}}}+{c}\)
\(\displaystyle\frac{{2}}{{9}}{x}^{{\frac{{9}}{{2}}}}+\frac{{1}}{{2}}\cdot\frac{{2}}{{5}}{x}^{{\frac{{5}}{{2}}}}+{c}\)
\(\displaystyle=\frac{{2}}{{9}}{x}^{{\frac{{9}}{{2}}}}+\frac{{1}}{{5}}{x}^{{\frac{{5}}{{2}}}}+{c}\)
\(\displaystyle\therefore\int\sqrt{{x}}{\left({x}^{{3}}+\frac{{x}}{{2}}\right)}{\left.{d}{x}\right.}=\frac{{2}}{{9}}{x}^{{\frac{{9}}{{2}}}}+\frac{{1}}{{5}}{x}^{{\frac{{5}}{{2}}}}+{c}\)
0

Relevant Questions

asked 2021-05-26
use the table of integrals to evaluate the following integral.
\(\int 3x \sqrt{6x-x^2}dx\)
asked 2021-05-28
Evaluate the following integral.
\(\int \frac{3x^{2}+\sqrt{x}}{\sqrt{x}}dx\)
asked 2021-05-29
Evaluate the following integral: \(\int \frac{x+3}{x-1}dx\)
asked 2021-05-23
Use a change of variables to evaluate the following integral.
\(\int-(\cos^{7}x-5\cos^{5}x-\cos x)\sin x dx\)
asked 2021-05-23
Find the indefinite integral \(\int \ln(\frac{x}{3})dx\) (a) using a table of integrals and (b) using the Integration by parts method.
asked 2021-05-27
Evaluate the following integral.
\(\int 2x^{3}+3x-2dx\)
asked 2021-05-11
Evaluate the following integral: \(\int \frac{(y-3)}{y^{2}-6y+1}\)
asked 2021-05-11
Evaluate the surface integral.
\(\int \int_s (x^2z +y^2z)dS\)
S is the hemisphere \(x^2 +y^2 + z^2 = 4, z \geq 0\)
asked 2021-05-23
Evaluate each of the following integrals.
\(\int\frac{e^{x}}{1+e^{x}}dx\)
asked 2021-05-17
the graph of g consists of two straight lines and a semicircle. use it to evaluate each integral

a)\(\int_0^{10}g(x)dx\)
b)\(\int_{10}^{30}g(x)dx\)
c)\(\int_0^{35}g(x)dx\)
...