New Notes Blog: `int_0^(pi/4)(sec^2 (t))dt` Evaluate the integral.

Saturday, 16 August 2014

$\displaystyle{\int_{{0}}^{{\frac{\pi}{{4}}}}}{\left({{\sec}}^{{2}}{\left({t}\right)}\right)}{\left.{d}{t}\right.}$ Evaluate the integral.

$\displaystyle{\int_{{0}}^{{\frac{\pi}{{4}}}}}{{\sec}}^{{2}}{\left({t}\right)}{\left.{d}{t}\right.}$

Take note that the derivative of tangent is d/(d theta) tan (theta)= sec^2 (theta).

So taking the integral of sec^2(t) result to:

$\displaystyle={\tan{{\left({t}\right)}}}{{\mid}_{{0}}^{{\frac{\pi}{{4}}}}}$

Plug-in the limits of integral as follows $\displaystyle{F}{\left({x}\right)}={\int_{{a}}^{{b}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={F}{\left({b}\right)}-{F}{\left({a}\right)}$ .

$\displaystyle={\tan{{\left(\frac{\pi}{{4}}\right)}}}-{\tan{{\left({0}\right)}}}$

$\displaystyle={1}-{0}$

$\displaystyle={1}$

Therefore, $\displaystyle{\int_{{0}}^{{\frac{\pi}{{4}}}}}{{\sec}}^{{2}}{\left({t}\right)}{\left.{d}{t}\right.}={1}$ .

$\displaystyle{\int_{{0}}^{{\frac{\pi}{{4}}}}}{{\sec}}^{{2}}{\left({t}\right)}{\left.{d}{t}\right.}$

Take note that the derivative of tangent is d/(d theta) tan (theta)= sec^2 (theta).

So taking the integral of sec^2(t) result to:

$\displaystyle={\tan{{\left({t}\right)}}}{{\mid}_{{0}}^{{\frac{\pi}{{4}}}}}$

Plug-in the limits of integral as follows $\displaystyle{F}{\left({x}\right)}={\int_{{a}}^{{b}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={F}{\left({b}\right)}-{F}{\left({a}\right)}$ .

$\displaystyle={\tan{{\left(\frac{\pi}{{4}}\right)}}}-{\tan{{\left({0}\right)}}}$

$\displaystyle={1}-{0}$

$\displaystyle={1}$

Therefore, $\displaystyle{\int_{{0}}^{{\frac{\pi}{{4}}}}}{{\sec}}^{{2}}{\left({t}\right)}{\left.{d}{t}\right.}={1}$ .

New Notes Blog

Saturday, 16 August 2014

$\displaystyle{\int_{{0}}^{{\frac{\pi}{{4}}}}}{\left({{\sec}}^{{2}}{\left({t}\right)}\right)}{\left.{d}{t}\right.}$ Evaluate the integral.

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In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?

Saturday, 16 August 2014

Evaluate the integral.

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Post a Comment

In &quot;By the Waters of Babylon,&quot; under the leadership of John, what do you think the Hill People will do with their society?

$\displaystyle{\int_{{0}}^{{\frac{\pi}{{4}}}}}{\left({{\sec}}^{{2}}{\left({t}\right)}\right)}{\left.{d}{t}\right.}$ Evaluate the integral.

In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?