Mister Exam

Integral of sqrt(a-bx^2) dx

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The solution

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$$\int\limits_{0}^{1} \sqrt{- b x^{2} + a}\, dx$$
Detail solution

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)

1. Now simplify:

2. Add the constant of integration:

  /                       //     __________        /                     __________\            \
|                        ||    /        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                            |
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\\                        x*\/ a                            otherwise /
$$\int \sqrt{- b x^{2} + a}\, dx = C + \begin{cases} \frac{x \sqrt{- b x^{2} + a}}{2} + \frac{a \log{\left(- 2 b x + 2 \sqrt{- b} \sqrt{- b x^{2} + a} \right)}}{2 \sqrt{- b}} & \text{for}\: b \neq 0 \\\sqrt{a} x & \text{otherwise} \end{cases}$$
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|  |    ___    /       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
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|  |           ___                    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  /
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$$\int\limits_{0}^{1} \begin{cases} \frac{i \sqrt{a} \sqrt{\frac{b x^{2}}{a} - 1}}{2} + \frac{i b x^{2}}{2 \sqrt{a} \sqrt{\frac{b x^{2}}{a} - 1}} - \frac{i \sqrt{a}}{2 \sqrt{\frac{\sqrt{b} x}{\sqrt{a}} - 1} \sqrt{\frac{\sqrt{b} x}{\sqrt{a}} + 1}} & \text{for}\: x^{2} \left|{\frac{b}{a}}\right| > 1 \\- \frac{3 b x^{2}}{2 \sqrt{a} \sqrt{- \frac{b x^{2}}{a} + 1}} + \frac{\sqrt{a}}{\sqrt{- \frac{b x^{2}}{a} + 1}} - \frac{b^{2} x^{4}}{2 a^{\frac{3}{2}} \left(- \frac{b x^{2}}{a} + 1\right)^{\frac{3}{2}}} + \frac{b x^{2}}{2 \sqrt{a} \left(- \frac{b x^{2}}{a} + 1\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx$$
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|  /             ___________
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|  |    ___    /       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
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|  |           ___                    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  /
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$$\int\limits_{0}^{1} \begin{cases} \frac{i \sqrt{a} \sqrt{\frac{b x^{2}}{a} - 1}}{2} + \frac{i b x^{2}}{2 \sqrt{a} \sqrt{\frac{b x^{2}}{a} - 1}} - \frac{i \sqrt{a}}{2 \sqrt{\frac{\sqrt{b} x}{\sqrt{a}} - 1} \sqrt{\frac{\sqrt{b} x}{\sqrt{a}} + 1}} & \text{for}\: x^{2} \left|{\frac{b}{a}}\right| > 1 \\- \frac{3 b x^{2}}{2 \sqrt{a} \sqrt{- \frac{b x^{2}}{a} + 1}} + \frac{\sqrt{a}}{\sqrt{- \frac{b x^{2}}{a} + 1}} - \frac{b^{2} x^{4}}{2 a^{\frac{3}{2}} \left(- \frac{b x^{2}}{a} + 1\right)^{\frac{3}{2}}} + \frac{b x^{2}}{2 \sqrt{a} \left(- \frac{b x^{2}}{a} + 1\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx$$

Use the examples entering the upper and lower limits of integration.