How do I evaluate the integral: $\int_{0}^{7 \sqrt{\frac{3}{2}}} \frac{d x}{\sqrt{49 - x^{2}}}$ $int_0^(7sqrt(3/2))dx / sqrt(49-x^2)$ ?

Question

How do I evaluate the integral: $\int_{0}^{7 \sqrt{\frac{3}{2}}} \frac{d x}{\sqrt{49 - x^{2}}}$ $int_0^(7sqrt(3/2))dx / sqrt(49-x^2)$ ?

1 Answer

Impact of this question

3163 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-01-27T18:08:56

First, recall the antiderivative formula:
$\int \frac{d x}{\sqrt{a^{2} - u^{2}}} = arcsin (\frac{u}{a}) + C$ $int dx/sqrt(a^2-u^2) = arcsin(u/a) + C$

Thus, $\int \frac{d x}{\sqrt{49 - x^{2}}} = \int \frac{d x}{\sqrt{7^{2} - x^{2}}}$ $int dx/sqrt(49-x^2) = int dx/sqrt(7^2-x^2)$

We have $\int \frac{d x}{\sqrt{7^{2} - x^{2}}} = arcsin (\frac{x}{7}) + C$ $int dx/sqrt(7^2-x^2) = arcsin(x/7) + C$

Substitute in the constraints we

$\int_{0}^{7 \sqrt{\frac{3}{2}}} \frac{d x}{\sqrt{7^{2} - x^{2}}} = arcsin ⎛ ⎜ ⎜ ⎝ \frac{7 \sqrt{\frac{3}{2}}}{7} ⎞ ⎟ ⎟ ⎠ - arcsin (0) = arcsin ⎛ ⎜ ⎜ ⎝ \frac{7 \sqrt{\frac{3}{2}}}{7} ⎞ ⎟ ⎟ ⎠$ $int_0^(7sqrt(3/2)) dx/sqrt(7^2-x^2) = arcsin((7sqrt(3/2))/7) - arcsin(0) = arcsin((7sqrt(3/2))/7)$

Please find the formula sheet below for reference:
$www.eeweb.com$

Science

Math

Humanities

... and beyond

How do I evaluate the integral: $\int_{0}^{7 \sqrt{\frac{3}{2}}} \frac{d x}{\sqrt{49 - x^{2}}}$ $int_0^(7sqrt(3/2))dx / sqrt(49-x^2)$ ?

1 Answer

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do I evaluate the integral: ∫7√320dx√49−x2int_0^(7sqrt(3/2))dx / sqrt(49-x^2)?

1 Answer

Related questions

Impact of this question

How do I evaluate the integral: $\int_{0}^{7 \sqrt{\frac{3}{2}}} \frac{d x}{\sqrt{49 - x^{2}}}$ $int_0^(7sqrt(3/2))dx / sqrt(49-x^2)$ ?