WEBVTT
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Use integration by parts to find the exact value of the integral of π₯ squared sin two π₯ ππ₯ between zero and π by four.
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Integration by parts uses the formula that the integral of π’π£ dash is equal to π’π£ minus the integral of π£π’ dash, where π’ dash is the differential of π’, and π£ dash is the differential of π£.
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Our first step is to split our initial expression π₯ squared sin two π₯ into π’ and π£ dash.
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We will let π’ equal π₯ squared and π£ dash equals sin two π₯.
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To work out an expression for π’ dash, we need to differentiate π₯ squared.
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π₯ squared differentiated is two π₯.
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To work out an expression for π£, we need to integrate sin two π₯, as integration is the opposite of differentiation.
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The integral of sin two π₯ is equal to negative cos two π₯ over two.
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Multiplying π₯ squared by negative cos two π₯ over two gives us negative π₯ squared cos two π₯ over two.
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Multiplying two π₯ by negative cos two π₯ over two gives us negative two π₯ cos two π₯ over two.
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As we have two negative signs, this could be simplified to negative π₯ squared cos two π₯ over two plus the integral of two π₯ cos two π₯ divided by two.
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We can also cancel the twos after the integration sign, by dividing the numerator and denominator by two.
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We now need to try and integrate π₯ cos two π₯.
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We can integrate this expression using parts once again.
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We will let π’ equal π₯ and π£ dash equal cos two π₯.
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Differentiating π₯ gives us one.
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Therefore, π’ dash is equal to one.
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Integrating cos two π₯ gives us sin two π₯ over two.
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Multiplying π’ and π£, π₯ and sin two π₯ over two, give us π₯ sin two π₯ over two.
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Multiplying π’ dash and π£ gives us sin two π₯ over two.
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We are now left with negative π₯ squared cos two π₯ over two plus π₯ sin two π₯ over two minus the integral of sin two π₯ over two.
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Integrating the third term, sin two π₯ over two, gives us negative cos two π₯ over four.
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Once again, our two negative sins can turn into a positive.
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This means that the integral of π₯ squared sin two π₯ is equal to negative π₯ squared cos two π₯ over two plus π₯ sin two π₯ over two plus cos two π₯ over four.
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Our final step is to substitute our two limits, π by four and zero, and subtract the two answers.
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Firstly, letβs substitute π₯ equals π by four.
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Before starting, itβs worth noting that our trigonometrical functions are cos two π₯ and sin two π₯.
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Therefore, we need to calculate cos of two π by four and sin of two π by four.
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Two π by four radians is the same as π by two radians.
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And π by two radians is equal to 90 degrees.
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We know from our trig graphs that cos of 90 is equal to zero.
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Therefore, cos of π by two radians must also equal zero.
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The sin of 90 degrees is equal to one.
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Therefore, the sin of π by two radians is also equal to one.
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As cos of π by two is equal to zero, we know that the first and third terms in our expression will be equal to zero when π₯ is equal to π by four.
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The only term that will give us a value is π₯ sin two π₯ over two.
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As π₯ is equal to π by four and sin of two π₯ is equal to one, this term gives us π by four multiplied by one divided by two.
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This is equal to π by eight, as π by four divided by two is equal to π by eight.
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When π₯ is equal to π by four, the integral of π₯ squared sin two π₯ equals π by eight.
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We also need to consider the lower limit when π₯ is equal to zero.
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Once again, from our trig graphs, we know that cos of zero is equal to one and sin of zero is equal to zero.
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This means that the middle term π₯ sin two π₯ over two will be equal to zero.
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It is possible that the first and third term will have nonzero values, as cos of two π₯ is equal to one.
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The first term negative π₯ squared cos two π₯ over two also has an π₯ term.
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And as π₯ is equal to zero, this whole term will also equal zero.
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Therefore, the only nonzero term is cos of two π₯ over four.
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We already know that cos two π₯ is equal to one.
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Therefore, cos two π₯ over four is equal to one-quarter.
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As we have worked out exact values for the upper and lower limit, we can now say that the integral of π₯ squared sin two π₯ between zero and π by four is π by eight minus one-quarter.