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- Improper integral
- Double Integral
- Triple Integral
- Curvilinear integral
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How to use it?
- Integral of d{x}:
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Integral of 8-2x^2
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Integral of (x^2+7x-5)cos2x
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Integral of ctg^2(x)dx
- Integral of ln(ax+b)/√(ax+b)
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Identical expressions
- ln(ax+b)/√(ax+b)
- ln(ax plus b) divide by √(ax plus b)
- lnax+b/√ax+b
- ln(ax+b) divide by √(ax+b)
- ln(ax+b)/√(ax+b)dx
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Similar expressions
- ln(ax-b)/√(ax+b)
- ln(ax+b)/√(ax-b)
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What you mean?
- log(a*x + b)/(sqrt(a*x + b))
- log(a*x + b)/(sqrt(a*x) + b)
- log(a*x + b)/(sqrt(a)*x + b)
- log(a*x + b)*(a*x + b)/(sqrt(x))
- log(a*x + b)/(sqrt(a*x + b))
- log(a*x + b)/(sqrt(a*x) + b)
- log(a*x + b)*(a*x + b)/(sqrt(x))
You entered:
ln(ax+b)/√(ax+b)
What you mean?
Integral of ln(ax+b)/√(ax+b) dx
The solution
You have entered
[src]
1 / | | log(a*x + b) | ------------ dx | _________ | \/ a*x + b | / 0
$$\int\limits_{0}^{1} \frac{\log{\left(a x + b \right)}}{\sqrt{a x + b}}\, dx$$
Integral(log(a*x + b)/(sqrt(a*x + b)), (x, 0, 1))
Detail solution
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Now simplify:
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Add the constant of integration:
PiecewiseRule(subfunctions=[(URule(u_var=_u, u_func=sqrt(a*x + b), constant=None, substep=ConstantTimesRule(constant=2/a, other=log(_u**2), substep=PartsRule(u=log(_u**2), dv=1, v_step=ConstantRule(constant=1, context=1, symbol=_u), second_step=ConstantRule(constant=2, context=2, symbol=_u), context=log(_u**2), symbol=_u), context=2*log(_u**2)/a, symbol=_u), context=log(a*x + b)/(sqrt(a*x + b)), symbol=x), Ne(a, 0)), (ConstantRule(constant=log(b)/sqrt(b), context=log(b)/sqrt(b), symbol=x), True)], context=log(a*x + b)/(sqrt(a*x + b)), symbol=x)
The answer is:
The answer (Indefinite)
[src]
// / _________ _________ \ \ / ||2*\- 2*\/ b + a*x + \/ b + a*x *log(b + a*x)/ | | ||---------------------------------------------- for a != 0| | log(a*x + b) || a | | ------------ dx = C + |< | | _________ || x*log(b) | | \/ a*x + b || -------- otherwise | | || ___ | / \\ \/ b /
$${{4\,\left({{\sqrt{a\,x+b}\,\log \left(a\,x+b\right)}\over{2}}-
\sqrt{a\,x+b}\right)}\over{a}}$$
The answer
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/ / ___ ___ \ / _______ _______ \ | 2*\- 2*\/ b + \/ b *log(b)/ 2*\- 2*\/ a + b + \/ a + b *log(a + b)/ |- ---------------------------- + ---------------------------------------- for And(a > -oo, a < oo, a != 0) | a a < | log(b) | ------ otherwise | ___ \ \/ b
$$\begin{cases} - \frac{2 \left(\sqrt{b} \log{\left(b \right)} - 2 \sqrt{b}\right)}{a} + \frac{2 \left(\sqrt{a + b} \log{\left(a + b \right)} - 2 \sqrt{a + b}\right)}{a} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\frac{\log{\left(b \right)}}{\sqrt{b}} & \text{otherwise} \end{cases}$$
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/ / ___ ___ \ / _______ _______ \ | 2*\- 2*\/ b + \/ b *log(b)/ 2*\- 2*\/ a + b + \/ a + b *log(a + b)/ |- ---------------------------- + ---------------------------------------- for And(a > -oo, a < oo, a != 0) | a a < | log(b) | ------ otherwise | ___ \ \/ b
$$\begin{cases} - \frac{2 \left(\sqrt{b} \log{\left(b \right)} - 2 \sqrt{b}\right)}{a} + \frac{2 \left(\sqrt{a + b} \log{\left(a + b \right)} - 2 \sqrt{a + b}\right)}{a} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\frac{\log{\left(b \right)}}{\sqrt{b}} & \text{otherwise} \end{cases}$$
Use the examples entering the upper and lower limits of integration.