Solve \int_{0}^{a} x \sqrt{a^{2}-x^{2}} d x - ScanMath

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Type your math problem... \int_{0}^{a} x \sqrt{a^{2}-x^{2}} d x

Question

Evaluate the integral

\frac{1}{3} a^{3}

Evaluate

\int_{0}^{a} x a^{2} - x^{2} d x

Simplify

\int_{0}^{a} x (a^{2} - x^{2})^{\frac{1}{2}} d x

Evaluate the integral

\int x (a^{2} - x^{2})^{\frac{1}{2}} d x

Use the substitution d x = - \frac{1}{2 x} d t to transform the integral

More Steps

\int x (a^{2} - x^{2})^{\frac{1}{2}} (- \frac{1}{2 x}) d t

Simplify

More Steps

\int - \frac{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}}{2} d t

Use the substitution t = - x^{2} to transform the integral

\int \frac{- ( a ^{2} + t ) ^{\frac{1}{2}}}{2} d t

Rewrite the expression

\int \frac{1}{2} (- (a^{2} + t)^{\frac{1}{2}}) d t

Use the property of integral \int k f (x) d x = k \int f (x) d x

\frac{1}{2} \times \int - (a^{2} + t)^{\frac{1}{2}} d t

Use the substitution d t = 1 d v to transform the integral

More Steps

\frac{1}{2} \times \int - (a^{2} + t)^{\frac{1}{2}} \times 1 d v

Any expression multiplied by 1 remains the same

\frac{1}{2} \times \int - (a^{2} + t)^{\frac{1}{2}} d v

Use the substitution v = a^{2} + t to transform the integral

\frac{1}{2} \times \int - v^{\frac{1}{2}} d v

Use the property of integral \int k f (x) d x = k \int f (x) d x

\frac{1}{2} (- 1) \times \int v^{\frac{1}{2}} d v

Multiplying or dividing an odd number of negative terms equals a negative

- \frac{1}{2} \times \int v^{\frac{1}{2}} d v

Use the property of integral \int x^{n} d x = \frac{x ^{n + 1}}{n + 1}

- \frac{1}{2} \times \frac{v ^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}

Simplify

More Steps

- \frac{1}{2} \times \frac{v ^{\frac{3}{2}}}{\frac{3}{2}}

Multiply the terms

- \frac{v ^{\frac{3}{2}}}{2 \times \frac{3}{2}}

Multiply the terms

More Steps

- \frac{v ^{\frac{3}{2}}}{3}

Substitute back

- \frac{( a ^{2} + t ) ^{\frac{3}{2}}}{3}

Substitute back

- \frac{( a ^{2} - x ^{2} ) ^{\frac{3}{2}}}{3}

Simplify

- \frac{1}{3} (a^{2} - x^{2})^{\frac{3}{2}}

Return the limits

(- \frac{1}{3} (a^{2} - x^{2})^{\frac{3}{2}})_{0}^{a}

Solution

More Steps

\frac{1}{3} a^{3}

Show Solution