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- Improper integral
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How to use it?
- Integral of d{x}:
- Integral of sqrt(a-bx^2)
- Integral of 3-sin(x)/(2cos(x)+3tan(x))
- Integral of 1/(x^2-8x-9)
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Integral of sin(4x+3)
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Identical expressions
- sqrt(a-bx^ two)
- square root of (a minus bx squared )
- square root of (a minus bx to the power of two)
- √(a-bx^2)
- sqrt(a-bx2)
- sqrta-bx2
- sqrt(a-bx²)
- sqrt(a-bx to the power of 2)
- sqrta-bx^2
- sqrt(a-bx^2)dx
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Similar expressions
- sqrt(a+bx^2)
Integral of sqrt(a-bx^2) dx
The solution
You have entered
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1 / | | __________ | / 2 | \/ a - b*x dx | / 0
$$\int\limits_{0}^{1} \sqrt{- b x^{2} + a}\, dx$$
Detail solution
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Now simplify:
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Add the constant of integration:
PiecewiseRule(subfunctions=[(SqrtQuadraticRule(a=a, b=0, c=-b, context=sqrt(a - b*x**2), symbol=x), Ne(-b, 0)), (ConstantRule(constant=sqrt(a), context=sqrt(a), symbol=x), True)], context=sqrt(a - b*x**2), symbol=x)
The answer is:
The answer (Indefinite)
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/ // __________ / __________\ \
| || / 2 | ____ / 2 | |
| __________ ||x*\/ a - b*x a*log\-2*b*x + 2*\/ -b *\/ a - b*x / |
| / 2 ||--------------- + -------------------------------------- for b != 0|
| \/ a - b*x dx = C + |< 2 ____ |
| || 2*\/ -b |
/ || |
|| ___ |
\\ x*\/ a otherwise /
$$\int \sqrt{- b x^{2} + a}\, dx = C + \begin{cases} \frac{x \sqrt{- b x^{2} + a}}{2} + \frac{a \log{\left(- 2 b x + 2 \sqrt{- b} \sqrt{- b x^{2} + a} \right)}}{2 \sqrt{- b}} & \text{for}\: b \neq 0 \\\sqrt{a} x & \text{otherwise} \end{cases}$$
The answer
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1 / | | / ___________ | | / 2 | | ___ / b*x | |I*\/ a * / -1 + ---- ___ 2 | | \/ a I*\/ a I*b*x 2 |b| | |------------------------ - ------------------------------------------ + ------------------------ for x *|-| > 1 | | 2 _____________ ______________ ___________ |a| | | / ___ / ___ / 2 | | / x*\/ b / x*\/ b ___ / b*x | | 2* / 1 + ------- * / -1 + ------- 2*\/ a * / -1 + ---- | < / ___ / ___ \/ a dx | | \/ \/ a \/ \/ a | | | | ___ 2 2 2 4 | | \/ a b*x 3*b*x b *x | | --------------- + --------------------- - ----------------------- - -------------------- otherwise | | __________ 3/2 __________ 3/2 | | / 2 / 2\ / 2 / 2\ | | / b*x ___ | b*x | ___ / b*x 3/2 | b*x | | | / 1 - ---- 2*\/ a *|1 - ----| 2*\/ a * / 1 - ---- 2*a *|1 - ----| | \ \/ a \ a / \/ a \ a / | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{i \sqrt{a} \sqrt{\frac{b x^{2}}{a} - 1}}{2} + \frac{i b x^{2}}{2 \sqrt{a} \sqrt{\frac{b x^{2}}{a} - 1}} - \frac{i \sqrt{a}}{2 \sqrt{\frac{\sqrt{b} x}{\sqrt{a}} - 1} \sqrt{\frac{\sqrt{b} x}{\sqrt{a}} + 1}} & \text{for}\: x^{2} \left|{\frac{b}{a}}\right| > 1 \\- \frac{3 b x^{2}}{2 \sqrt{a} \sqrt{- \frac{b x^{2}}{a} + 1}} + \frac{\sqrt{a}}{\sqrt{- \frac{b x^{2}}{a} + 1}} - \frac{b^{2} x^{4}}{2 a^{\frac{3}{2}} \left(- \frac{b x^{2}}{a} + 1\right)^{\frac{3}{2}}} + \frac{b x^{2}}{2 \sqrt{a} \left(- \frac{b x^{2}}{a} + 1\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx$$
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1 / | | / ___________ | | / 2 | | ___ / b*x | |I*\/ a * / -1 + ---- ___ 2 | | \/ a I*\/ a I*b*x 2 |b| | |------------------------ - ------------------------------------------ + ------------------------ for x *|-| > 1 | | 2 _____________ ______________ ___________ |a| | | / ___ / ___ / 2 | | / x*\/ b / x*\/ b ___ / b*x | | 2* / 1 + ------- * / -1 + ------- 2*\/ a * / -1 + ---- | < / ___ / ___ \/ a dx | | \/ \/ a \/ \/ a | | | | ___ 2 2 2 4 | | \/ a b*x 3*b*x b *x | | --------------- + --------------------- - ----------------------- - -------------------- otherwise | | __________ 3/2 __________ 3/2 | | / 2 / 2\ / 2 / 2\ | | / b*x ___ | b*x | ___ / b*x 3/2 | b*x | | | / 1 - ---- 2*\/ a *|1 - ----| 2*\/ a * / 1 - ---- 2*a *|1 - ----| | \ \/ a \ a / \/ a \ a / | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{i \sqrt{a} \sqrt{\frac{b x^{2}}{a} - 1}}{2} + \frac{i b x^{2}}{2 \sqrt{a} \sqrt{\frac{b x^{2}}{a} - 1}} - \frac{i \sqrt{a}}{2 \sqrt{\frac{\sqrt{b} x}{\sqrt{a}} - 1} \sqrt{\frac{\sqrt{b} x}{\sqrt{a}} + 1}} & \text{for}\: x^{2} \left|{\frac{b}{a}}\right| > 1 \\- \frac{3 b x^{2}}{2 \sqrt{a} \sqrt{- \frac{b x^{2}}{a} + 1}} + \frac{\sqrt{a}}{\sqrt{- \frac{b x^{2}}{a} + 1}} - \frac{b^{2} x^{4}}{2 a^{\frac{3}{2}} \left(- \frac{b x^{2}}{a} + 1\right)^{\frac{3}{2}}} + \frac{b x^{2}}{2 \sqrt{a} \left(- \frac{b x^{2}}{a} + 1\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx$$
Use the examples entering the upper and lower limits of integration.