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Identical expressions
- sqrt(four - five *x)*sinx
- square root of (4 minus 5 multiply by x) multiply by sinus of x
- square root of (four minus five multiply by x) multiply by sinus of x
- √(4-5*x)*sinx
- sqrt(4-5x)sinx
- sqrt4-5xsinx
- sqrt(4-5*x)*sinxdx
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Similar expressions
- sqrt(4+5*x)*sinx
Integral of sqrt(4-5*x)*sinx dx
The solution
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[src]
pi -- 2 / | | _________ | \/ 4 - 5*x *sin(x) dx | / 0
$$\int\limits_{0}^{\frac{\pi}{2}} \sqrt{4 - 5 x} \sin{\left(x \right)}\, dx$$
Integral(sqrt(4 - 5*x)*sin(x), (x, 0, pi/2))
The answer (Indefinite)
[src]
/ \
| _ / | 2 \ _ / | 2 \|
| ____ 3/2 |_ | 1/4, 3/4 | -(4 - 5*x) | ____ 5/2 |_ | 3/4, 5/4 | -(4 - 5*x) ||
| \/ 10 *(4 - 5*x) *Gamma(1/4)*Gamma(3/4)* | | | ------------|*sin(4/5) \/ 10 *(4 - 5*x) *cos(4/5)*Gamma(3/4)*Gamma(5/4)* | | | ------------|| / / ____ _________\ / ____ _________\ \
____ ____ | 2 3 \1/2, 5/4, 7/4 | 100 / 2 3 \3/2, 7/4, 9/4 | 100 /| ____ ____ | |\/ 10 *\/ 4 - 5*x | |\/ 10 *\/ 4 - 5*x | |
/ 2*\/ 10 *\/ pi *|- -------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------| \/ 10 *\/ pi *(4 - 5*x)*|cos(4/5)*S|------------------| - C|------------------|*sin(4/5)|
| | ____ ____ | | | ____ | | ____ | |
| _________ \ 80*\/ pi *Gamma(5/4)*Gamma(7/4) 400*\/ pi *Gamma(7/4)*Gamma(9/4) / \ \ 5*\/ pi / \ 5*\/ pi / /
| \/ 4 - 5*x *sin(x) dx = C - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -----------------------------------------------------------------------------------------
| 5 5
/
$$\int \sqrt{4 - 5 x} \sin{\left(x \right)}\, dx = C + \frac{\sqrt{10} \sqrt{\pi} \left(4 - 5 x\right) \left(- \sin{\left(\frac{4}{5} \right)} C\left(\frac{\sqrt{10} \sqrt{4 - 5 x}}{5 \sqrt{\pi}}\right) + \cos{\left(\frac{4}{5} \right)} S\left(\frac{\sqrt{10} \sqrt{4 - 5 x}}{5 \sqrt{\pi}}\right)\right)}{5} - \frac{2 \sqrt{10} \sqrt{\pi} \left(\frac{\sqrt{10} \left(4 - 5 x\right)^{\frac{5}{2}} \cos{\left(\frac{4}{5} \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {- \frac{\left(4 - 5 x\right)^{2}}{100}} \right)}}{400 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)} - \frac{\sqrt{10} \left(4 - 5 x\right)^{\frac{3}{2}} \sin{\left(\frac{4}{5} \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {- \frac{\left(4 - 5 x\right)^{2}}{100}} \right)}}{80 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)}\right)}{5}$$
The graph
The answer
[src]
/ \
| / | 2 \ / | 2 \|
| 3/2 _ | | / 5*pi\ | 5/2 _ | | / 5*pi\ ||
| ____ / 5*pi\ |_ | 1/4, 3/4 | -|4 - ----| | ____ / 5*pi\ |_ | 3/4, 5/4 | -|4 - ----| || / _ _ \ / / __________\ / __________\ \
| \/ 10 *|4 - ----| *Gamma(1/4)*Gamma(3/4)* | | | \ 2 / |*sin(4/5) \/ 10 *|4 - ----| *cos(4/5)*Gamma(3/4)*Gamma(5/4)* | | | \ 2 / || | ____ |_ / 1/4, 3/4 | \ ____ |_ / 3/4, 5/4 | \| | | ____ / 5*pi | | ____ / 5*pi | |
/ / ____\ / ____\ \ | \ 2 / 2 3 |1/2, 5/4, 7/4 | -------------| \ 2 / 2 3 |3/2, 7/4, 9/4 | -------------|| | \/ 10 *Gamma(1/4)*Gamma(3/4)* | | | -4/25|*sin(4/5) 2*\/ 10 *cos(4/5)*Gamma(3/4)*Gamma(5/4)* | | | -4/25|| | |\/ 10 * / 4 - ---- | |\/ 10 * / 4 - ---- | |
____ ____ | |2*\/ 10 | |2*\/ 10 | | ____ ____ | \ | 100 / \ | 100 /| ____ ____ | 2 3 \1/2, 5/4, 7/4 | / 2 3 \3/2, 7/4, 9/4 | /| ____ ____ / 5*pi\ | | \/ 2 | | \/ 2 | |
4*\/ 10 *\/ pi *|cos(4/5)*S|--------| - C|--------|*sin(4/5)| 2*\/ 10 *\/ pi *|- ---------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------| 2*\/ 10 *\/ pi *|- ------------------------------------------------------------------ + --------------------------------------------------------------------| \/ 10 *\/ pi *|4 - ----|*|cos(4/5)*S|---------------------| - C|---------------------|*sin(4/5)|
| | ____| | ____| | | ____ ____ | | ____ ____ | \ 2 / | | ____ | | ____ | |
\ \5*\/ pi / \5*\/ pi / / \ 80*\/ pi *Gamma(5/4)*Gamma(7/4) 400*\/ pi *Gamma(7/4)*Gamma(9/4) / \ 10*\/ pi *Gamma(5/4)*Gamma(7/4) 25*\/ pi *Gamma(7/4)*Gamma(9/4) / \ \ 5*\/ pi / \ 5*\/ pi / /
- ------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------
5 5 5 5
$$\frac{2 \sqrt{10} \sqrt{\pi} \left(- \frac{\sqrt{10} \sin{\left(\frac{4}{5} \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {- \frac{4}{25}} \right)}}{10 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{2 \sqrt{10} \cos{\left(\frac{4}{5} \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {- \frac{4}{25}} \right)}}{25 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}\right)}{5} - \frac{2 \sqrt{10} \sqrt{\pi} \left(- \frac{\sqrt{10} \left(4 - \frac{5 \pi}{2}\right)^{\frac{3}{2}} \sin{\left(\frac{4}{5} \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {- \frac{\left(4 - \frac{5 \pi}{2}\right)^{2}}{100}} \right)}}{80 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{\sqrt{10} \left(4 - \frac{5 \pi}{2}\right)^{\frac{5}{2}} \cos{\left(\frac{4}{5} \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {- \frac{\left(4 - \frac{5 \pi}{2}\right)^{2}}{100}} \right)}}{400 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}\right)}{5} - \frac{4 \sqrt{10} \sqrt{\pi} \left(- \sin{\left(\frac{4}{5} \right)} C\left(\frac{2 \sqrt{10}}{5 \sqrt{\pi}}\right) + \cos{\left(\frac{4}{5} \right)} S\left(\frac{2 \sqrt{10}}{5 \sqrt{\pi}}\right)\right)}{5} + \frac{\sqrt{10} \sqrt{\pi} \left(4 - \frac{5 \pi}{2}\right) \left(- \sin{\left(\frac{4}{5} \right)} C\left(\frac{\sqrt{10} \sqrt{4 - \frac{5 \pi}{2}}}{5 \sqrt{\pi}}\right) + \cos{\left(\frac{4}{5} \right)} S\left(\frac{\sqrt{10} \sqrt{4 - \frac{5 \pi}{2}}}{5 \sqrt{\pi}}\right)\right)}{5}$$
=
=
/ \
| / | 2 \ / | 2 \|
| 3/2 _ | | / 5*pi\ | 5/2 _ | | / 5*pi\ ||
| ____ / 5*pi\ |_ | 1/4, 3/4 | -|4 - ----| | ____ / 5*pi\ |_ | 3/4, 5/4 | -|4 - ----| || / _ _ \ / / __________\ / __________\ \
| \/ 10 *|4 - ----| *Gamma(1/4)*Gamma(3/4)* | | | \ 2 / |*sin(4/5) \/ 10 *|4 - ----| *cos(4/5)*Gamma(3/4)*Gamma(5/4)* | | | \ 2 / || | ____ |_ / 1/4, 3/4 | \ ____ |_ / 3/4, 5/4 | \| | | ____ / 5*pi | | ____ / 5*pi | |
/ / ____\ / ____\ \ | \ 2 / 2 3 |1/2, 5/4, 7/4 | -------------| \ 2 / 2 3 |3/2, 7/4, 9/4 | -------------|| | \/ 10 *Gamma(1/4)*Gamma(3/4)* | | | -4/25|*sin(4/5) 2*\/ 10 *cos(4/5)*Gamma(3/4)*Gamma(5/4)* | | | -4/25|| | |\/ 10 * / 4 - ---- | |\/ 10 * / 4 - ---- | |
____ ____ | |2*\/ 10 | |2*\/ 10 | | ____ ____ | \ | 100 / \ | 100 /| ____ ____ | 2 3 \1/2, 5/4, 7/4 | / 2 3 \3/2, 7/4, 9/4 | /| ____ ____ / 5*pi\ | | \/ 2 | | \/ 2 | |
4*\/ 10 *\/ pi *|cos(4/5)*S|--------| - C|--------|*sin(4/5)| 2*\/ 10 *\/ pi *|- ---------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------| 2*\/ 10 *\/ pi *|- ------------------------------------------------------------------ + --------------------------------------------------------------------| \/ 10 *\/ pi *|4 - ----|*|cos(4/5)*S|---------------------| - C|---------------------|*sin(4/5)|
| | ____| | ____| | | ____ ____ | | ____ ____ | \ 2 / | | ____ | | ____ | |
\ \5*\/ pi / \5*\/ pi / / \ 80*\/ pi *Gamma(5/4)*Gamma(7/4) 400*\/ pi *Gamma(7/4)*Gamma(9/4) / \ 10*\/ pi *Gamma(5/4)*Gamma(7/4) 25*\/ pi *Gamma(7/4)*Gamma(9/4) / \ \ 5*\/ pi / \ 5*\/ pi / /
- ------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------
5 5 5 5
$$\frac{2 \sqrt{10} \sqrt{\pi} \left(- \frac{\sqrt{10} \sin{\left(\frac{4}{5} \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {- \frac{4}{25}} \right)}}{10 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{2 \sqrt{10} \cos{\left(\frac{4}{5} \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {- \frac{4}{25}} \right)}}{25 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}\right)}{5} - \frac{2 \sqrt{10} \sqrt{\pi} \left(- \frac{\sqrt{10} \left(4 - \frac{5 \pi}{2}\right)^{\frac{3}{2}} \sin{\left(\frac{4}{5} \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {- \frac{\left(4 - \frac{5 \pi}{2}\right)^{2}}{100}} \right)}}{80 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{\sqrt{10} \left(4 - \frac{5 \pi}{2}\right)^{\frac{5}{2}} \cos{\left(\frac{4}{5} \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {- \frac{\left(4 - \frac{5 \pi}{2}\right)^{2}}{100}} \right)}}{400 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}\right)}{5} - \frac{4 \sqrt{10} \sqrt{\pi} \left(- \sin{\left(\frac{4}{5} \right)} C\left(\frac{2 \sqrt{10}}{5 \sqrt{\pi}}\right) + \cos{\left(\frac{4}{5} \right)} S\left(\frac{2 \sqrt{10}}{5 \sqrt{\pi}}\right)\right)}{5} + \frac{\sqrt{10} \sqrt{\pi} \left(4 - \frac{5 \pi}{2}\right) \left(- \sin{\left(\frac{4}{5} \right)} C\left(\frac{\sqrt{10} \sqrt{4 - \frac{5 \pi}{2}}}{5 \sqrt{\pi}}\right) + \cos{\left(\frac{4}{5} \right)} S\left(\frac{\sqrt{10} \sqrt{4 - \frac{5 \pi}{2}}}{5 \sqrt{\pi}}\right)\right)}{5}$$
-4*sqrt(10)*sqrt(pi)*(cos(4/5)*fresnels(2*sqrt(10)/(5*sqrt(pi))) - fresnelc(2*sqrt(10)/(5*sqrt(pi)))*sin(4/5))/5 - 2*sqrt(10)*sqrt(pi)*(-sqrt(10)*(4 - 5*pi/2)^(3/2)*gamma(1/4)*gamma(3/4)*hyper((1/4, 3/4), (1/2, 5/4, 7/4), -(4 - 5*pi/2)^2/100)*sin(4/5)/(80*sqrt(pi)*gamma(5/4)*gamma(7/4)) + sqrt(10)*(4 - 5*pi/2)^(5/2)*cos(4/5)*gamma(3/4)*gamma(5/4)*hyper((3/4, 5/4), (3/2, 7/4, 9/4), -(4 - 5*pi/2)^2/100)/(400*sqrt(pi)*gamma(7/4)*gamma(9/4)))/5 + 2*sqrt(10)*sqrt(pi)*(-sqrt(10)*gamma(1/4)*gamma(3/4)*hyper((1/4, 3/4), (1/2, 5/4, 7/4), -4/25)*sin(4/5)/(10*sqrt(pi)*gamma(5/4)*gamma(7/4)) + 2*sqrt(10)*cos(4/5)*gamma(3/4)*gamma(5/4)*hyper((3/4, 5/4), (3/2, 7/4, 9/4), -4/25)/(25*sqrt(pi)*gamma(7/4)*gamma(9/4)))/5 + sqrt(10)*sqrt(pi)*(4 - 5*pi/2)*(cos(4/5)*fresnels(sqrt(10)*sqrt(4 - 5*pi/2)/(5*sqrt(pi))) - fresnelc(sqrt(10)*sqrt(4 - 5*pi/2)/(5*sqrt(pi)))*sin(4/5))/5
Numerical answer
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(0.327486816197669 + 0.9423047673792j)
(0.327486816197669 + 0.9423047673792j)
Use the examples entering the upper and lower limits of integration.