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How to use it?
- Integral of d{x}:
- Integral of 1/(1+x^5)
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Integral of 3*exp(-3*x)
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Integral of 2*x/(1+x)
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Integral of (x*x)*exp(-x/2)
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Identical expressions
- exp(-ax^ two +bx)
- exponent of ( minus ax squared plus bx)
- exponent of ( minus ax to the power of two plus bx)
- exp(-ax2+bx)
- exp-ax2+bx
- exp(-ax²+bx)
- exp(-ax to the power of 2+bx)
- exp-ax^2+bx
- exp(-ax^2+bx)dx
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Similar expressions
- exp(ax^2+bx)
- exp(-ax^2-bx)
Integral of exp(-ax^2+bx) dx
The solution
You have entered
[src]
1 / | | 2 | -a*x + b*x | e dx | / 0
$$\int\limits_{0}^{1} e^{- a x^{2} + b x}\, dx$$
Integral(exp((-a)*x^2 + b*x), (x, 0, 1))
Detail solution
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Now simplify:
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Add the constant of integration:
ErfRule(a=-a, b=b, c=0, context=exp((-a)*x**2 + b*x), symbol=x)
The answer is:
The answer (Indefinite)
[src]
2
b
_____ ---
/ ____ / -1 /b - 2*a*x\ 4*a
| \/ pi * / --- *erfi|---------|*e
| 2 \/ a | ____|
| -a*x + b*x \ 2*\/ -a /
| e dx = C + -------------------------------------
| 2
/
$$\int e^{- a x^{2} + b x}\, dx = C + \frac{\sqrt{\pi} \sqrt{- \frac{1}{a}} e^{\frac{b^{2}}{4 a}} \operatorname{erfi}{\left(\frac{- 2 a x + b}{2 \sqrt{- a}} \right)}}{2}$$
The answer
[src]
2 2
b b
_____ --- _____ ---
____ / -1 /b - 2*a \ 4*a ____ / -1 / b \ 4*a
\/ pi * / --- *erfi|--------|*e \/ pi * / --- *erfi|--------|*e
\/ a | ____| \/ a | ____|
\2*\/ -a / \2*\/ -a /
------------------------------------ - ------------------------------------
2 2
$$- \frac{\sqrt{\pi} \sqrt{- \frac{1}{a}} e^{\frac{b^{2}}{4 a}} \operatorname{erfi}{\left(\frac{b}{2 \sqrt{- a}} \right)}}{2} + \frac{\sqrt{\pi} \sqrt{- \frac{1}{a}} e^{\frac{b^{2}}{4 a}} \operatorname{erfi}{\left(\frac{- 2 a + b}{2 \sqrt{- a}} \right)}}{2}$$
=
=
2 2
b b
_____ --- _____ ---
____ / -1 /b - 2*a \ 4*a ____ / -1 / b \ 4*a
\/ pi * / --- *erfi|--------|*e \/ pi * / --- *erfi|--------|*e
\/ a | ____| \/ a | ____|
\2*\/ -a / \2*\/ -a /
------------------------------------ - ------------------------------------
2 2
$$- \frac{\sqrt{\pi} \sqrt{- \frac{1}{a}} e^{\frac{b^{2}}{4 a}} \operatorname{erfi}{\left(\frac{b}{2 \sqrt{- a}} \right)}}{2} + \frac{\sqrt{\pi} \sqrt{- \frac{1}{a}} e^{\frac{b^{2}}{4 a}} \operatorname{erfi}{\left(\frac{- 2 a + b}{2 \sqrt{- a}} \right)}}{2}$$
sqrt(pi)*sqrt(-1/a)*erfi((b - 2*a)/(2*sqrt(-a)))*exp(b^2/(4*a))/2 - sqrt(pi)*sqrt(-1/a)*erfi(b/(2*sqrt(-a)))*exp(b^2/(4*a))/2
Use the examples entering the upper and lower limits of integration.