Mister Exam
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- Improper integral
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How to use it?
- Integral of d{x}:
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Integral of 1/5x
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Integral of (sec(x))^3
- Integral of lnx/sqrtx
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Integral of x^2*e^(x^3)
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Identical expressions
- √2xsin(x/ four)
- √2x sinus of (x divide by 4)
- √2x sinus of (x divide by four)
- √2xsinx/4
- √2xsin(x divide by 4)
- √2xsin(x/4)dx
Integral of √2xsin(x/4) dx
The solution
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pi / | | _____ /x\ | \/ 2*x *sin|-| dx | \4/ | / -pi
$$\int\limits_{- \pi}^{\pi} \sqrt{2 x} \sin{\left(\frac{x}{4} \right)}\, dx$$
Integral(sqrt(2*x)*sin(x/4), (x, -pi, pi))
The answer (Indefinite)
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___ ___
\/ 2 *\/ x
/
/ |
| | ____ / 2\
| _____ /x\ | / 2 |u |
| \/ 2*x *sin|-| dx = C + | \/ u *u*sin|--| du
| \4/ | \8 /
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/ /
$$\int \sqrt{2 x} \sin{\left(\frac{x}{4} \right)}\, dx = C + \int\limits^{\sqrt{2} \sqrt{x}} u \sqrt{u^{2}} \sin{\left(\frac{u^{2}}{8} \right)}\, du$$
The graph
The answer
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/ / ___\ \ / / ___\ \
| ___ ____ |\/ 2 | | | ___ ____ |\/ 2 | |
| ___ ____ 5*\/ 2 *\/ pi *C|-----|*Gamma(5/4)| | ___ ____ 5*I*\/ 2 *\/ pi *C|-----|*Gamma(5/4)|
___ | 5*\/ 2 *\/ pi *Gamma(5/4) \ 2 / | ___ | 5*I*\/ 2 *\/ pi *Gamma(5/4) \ 2 / |
\/ 2 *|- ------------------------- + ----------------------------------| - \/ 2 *|- --------------------------- + ------------------------------------|
\ 2*Gamma(9/4) Gamma(9/4) / \ 2*Gamma(9/4) Gamma(9/4) /
$$\sqrt{2} \left(- \frac{5 \sqrt{2} \sqrt{\pi} \Gamma\left(\frac{5}{4}\right)}{2 \Gamma\left(\frac{9}{4}\right)} + \frac{5 \sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2}}{2}\right) \Gamma\left(\frac{5}{4}\right)}{\Gamma\left(\frac{9}{4}\right)}\right) - \sqrt{2} \left(- \frac{5 \sqrt{2} i \sqrt{\pi} \Gamma\left(\frac{5}{4}\right)}{2 \Gamma\left(\frac{9}{4}\right)} + \frac{5 \sqrt{2} i \sqrt{\pi} C\left(\frac{\sqrt{2}}{2}\right) \Gamma\left(\frac{5}{4}\right)}{\Gamma\left(\frac{9}{4}\right)}\right)$$
=
=
/ / ___\ \ / / ___\ \
| ___ ____ |\/ 2 | | | ___ ____ |\/ 2 | |
| ___ ____ 5*\/ 2 *\/ pi *C|-----|*Gamma(5/4)| | ___ ____ 5*I*\/ 2 *\/ pi *C|-----|*Gamma(5/4)|
___ | 5*\/ 2 *\/ pi *Gamma(5/4) \ 2 / | ___ | 5*I*\/ 2 *\/ pi *Gamma(5/4) \ 2 / |
\/ 2 *|- ------------------------- + ----------------------------------| - \/ 2 *|- --------------------------- + ------------------------------------|
\ 2*Gamma(9/4) Gamma(9/4) / \ 2*Gamma(9/4) Gamma(9/4) /
$$\sqrt{2} \left(- \frac{5 \sqrt{2} \sqrt{\pi} \Gamma\left(\frac{5}{4}\right)}{2 \Gamma\left(\frac{9}{4}\right)} + \frac{5 \sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2}}{2}\right) \Gamma\left(\frac{5}{4}\right)}{\Gamma\left(\frac{9}{4}\right)}\right) - \sqrt{2} \left(- \frac{5 \sqrt{2} i \sqrt{\pi} \Gamma\left(\frac{5}{4}\right)}{2 \Gamma\left(\frac{9}{4}\right)} + \frac{5 \sqrt{2} i \sqrt{\pi} C\left(\frac{\sqrt{2}}{2}\right) \Gamma\left(\frac{5}{4}\right)}{\Gamma\left(\frac{9}{4}\right)}\right)$$
sqrt(2)*(-5*sqrt(2)*sqrt(pi)*gamma(5/4)/(2*gamma(9/4)) + 5*sqrt(2)*sqrt(pi)*fresnelc(sqrt(2)/2)*gamma(5/4)/gamma(9/4)) - sqrt(2)*(-5*i*sqrt(2)*sqrt(pi)*gamma(5/4)/(2*gamma(9/4)) + 5*i*sqrt(2)*sqrt(pi)*fresnelc(sqrt(2)/2)*gamma(5/4)/gamma(9/4))
Use the examples entering the upper and lower limits of integration.