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Definite Integrals of Trig FunctionsDefinite Integrals of Trig Functions
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Question 1 of 2
1. Question
Find the integral$$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\;\text{cos}x\;dx$$Hint
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Integrals of Trigonometric Functions
`int \text(cos)=\text(sin)``int \text(sin)=-\text(cos)``int \text(sec)^2=\text(tan)`First, integrate the trigonometric function$$\int_{\color{#00880A}{\frac{\pi}{4}}}^{\color{#9a00c7}{\frac{\pi}{2}}}\;\text{cos}\;x\;dx$$ `=` $$\bigg[\text{sin}\;x\bigg]_{\color{#00880A}{\frac{\pi}{4}}}^{\color{#9a00c7}{\frac{\pi}{2}}}$$ Integrate `\text(cos)x` Finally, get the difference of the upper and lower limits substituted to the integral as `x`.$$\bigg[\text{sin}\;x\bigg]_{\color{#00880A}{\frac{\pi}{4}}}^{\color{#9a00c7}{\frac{\pi}{2}}}$$ `=` $$\sin{\color{#9a00c7}{\frac{\pi}{2}}}-\sin{\color{#00880A}{\frac{\pi}{4}}}$$ Substitute the limits `=` `1-1/(sqrt2)` Evaluate `1-1/(sqrt2)` -
Question 2 of 2
2. Question
Find the integral$$\int_{0}^{\frac{\pi}{3}}\;3\;\text{sin}\frac{x}{2}\;dx$$Hint
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Integrals of Trigonometric Functions
`int \text(cos)=\text(sin)``int \text(sin)=-\text(cos)``int \text(sec)^2=\text(tan)`First, integrate the trigonometric function$$\int_{\color{#00880A}{0}}^{\color{#9a00c7}{\frac{\pi}{3}}}\;\text{sin}\;\frac{x}{2}\;dx$$ `=` $$\bigg[3\left(-\text{cos}\;\frac{x}{2}\right)\bigg]_{\color{#00880A}{0}}^{\color{#9a00c7}{\frac{\pi}{3}}}$$ Integrate `3\text(sin) x/2` `=` $$\bigg[-6\text{cos}\;\frac{x}{2}\bigg]_{\color{#00880A}{0}}^{\color{#9a00c7}{\frac{\pi}{3}}}$$ Simplify Finally, get the difference of the upper and lower limits substituted to the integral as `x`.$$\bigg[-6\text{cos}\;\frac{x}{2}\bigg]_{\color{#00880A}{0}}^{\color{#9a00c7}{\frac{\pi}{3}}}$$ `=` $$-6\;\cos{\frac{\color{#9a00c7}{\frac{\pi}{3}}}{2}}-[-6\;\cos{\frac{\color{#00880A}{0}}{2}}]$$ Substitute the limits `=` `-6 \text(cos)(pi/6)+(6*1)` Evaluate `=` `-6*(sqrt3)/2+6` `\text(cos) pi/6=(sqrt3)/2` `=` `-3sqrt3+6` Simplify `-3sqrt3+6`
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