New Notes Blog: `int_0^(pi/3) (sin(theta) + sin(theta)tan^2(theta))/(sec^2(theta)) d theta` Evaluate the integral

Saturday, 8 November 2014

$\displaystyle{\int_{{0}}^{{\frac{\pi}{{3}}}}}\frac{{{\sin{{\left(\theta\right)}}}+{\sin{{\left(\theta\right)}}}{{\tan}}^{{2}}{\left(\theta\right)}}}{{{{\sec}}^{{2}}{\left(\theta\right)}}}{d}\theta$ Evaluate the integral

Before evaluating this integral, simplify the expression in the integral using trigonometric identities. The following Pythagorean identity will be useful:

$\displaystyle{{\tan}}^{{2}}{\left(\theta\right)}+{1}={{\sec}}^{{2}}{\left(\theta\right)}$

Start by factoring out $\displaystyle{\sin{{\left(\theta\right)}}}$ from the numerator of the fraction:

$\displaystyle\frac{{{\sin{{\left(\theta\right)}}}+{\sin{{\left(\theta\right)}}}{{\tan}}^{{2}}{\left(\theta\right)}}}{{{{\sec}}^{{2}}{\left(\theta\right)}}}=\frac{{{\sin{{\left(\theta\right)}}}{\left({1}+{{\tan}}^{{2}}{\left(\theta\right)}\right)}}}{{{{\sec}}^{{2}}{\left(\theta\right)}}}$

Since $\displaystyle{1}+{{\tan}}^{{2}}{\left(\theta\right)}={{\sec}}^{{2}}{\left(\theta\right)}$ , the parenthesis in the numerator cancels, so the fraction equals $\displaystyle{\sin{{\left(\theta\right)}}}$ .

Then the given integral becomes

$\displaystyle{\int_{{0}}^{{\frac{\pi}{{3}}}}}{\sin{{\left(\theta\right)}}}{d}{\left(\theta\right)}=\ldots$

Before evaluating this integral, simplify the expression in the integral using trigonometric identities. The following Pythagorean identity will be useful:

$\displaystyle{{\tan}}^{{2}}{\left(\theta\right)}+{1}={{\sec}}^{{2}}{\left(\theta\right)}$

Start by factoring out $\displaystyle{\sin{{\left(\theta\right)}}}$ from the numerator of the fraction:

Then the given integral becomes

$\displaystyle{\int_{{0}}^{{\frac{\pi}{{3}}}}}{\sin{{\left(\theta\right)}}}{d}{\left(\theta\right)}=-{\cos{{\left(\theta\right)}}}{{\mid}_{{0}}^{{\frac{\pi}{{3}}}}}=-{\left({\cos{{\left(\frac{\pi}{{3}}\right)}}}-{\cos{{\left({0}\right)}}}\right)}=-{\left(\frac{{1}}{{2}}-{1}\right)}=\frac{{1}}{{2}}$

The given integral equals $\displaystyle\frac{{1}}{{2}}$ .

New Notes Blog

Saturday, 8 November 2014

$\displaystyle{\int_{{0}}^{{\frac{\pi}{{3}}}}}\frac{{{\sin{{\left(\theta\right)}}}+{\sin{{\left(\theta\right)}}}{{\tan}}^{{2}}{\left(\theta\right)}}}{{{{\sec}}^{{2}}{\left(\theta\right)}}}{d}\theta$ 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, 8 November 2014

Evaluate the integral

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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?

$\displaystyle{\int_{{0}}^{{\frac{\pi}{{3}}}}}\frac{{{\sin{{\left(\theta\right)}}}+{\sin{{\left(\theta\right)}}}{{\tan}}^{{2}}{\left(\theta\right)}}}{{{{\sec}}^{{2}}{\left(\theta\right)}}}{d}\theta$ 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?