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Essential Calculus: Early Transcendentals
Found in: Page 340
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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Short Answer

Evaluate the integral\(\int {x*{\mathop{\rm Sin}\nolimits} ({x^2})*{\mathop{\rm Cos}\nolimits} (3{x^2}).dx} \)

The Value of integral is \( - \frac{1}{2}{\mathop{\rm Cos}\nolimits} {x^2} + \frac{3}{2}{{\mathop{\rm Cos}\nolimits} ^2}{x^2} + c\)

See the step by step solution

Step by Step Solution

Step 1-

Apply the Formula of \({\mathop{\rm Cos}\nolimits} 3x\)

We know,\({\mathop{\rm Cos}\nolimits} 3x = 4{{\mathop{\rm Cos}\nolimits} ^3}x - 3{\mathop{\rm Cos}\nolimits} x\)

Therefore,\(\int {x*{\mathop{\rm Sin}\nolimits} {x^2}} *{\mathop{\rm Cos}\nolimits} 3{x^2}.dx = \int {x*{\mathop{\rm Sin}\nolimits} {x^2}(4{{{\mathop{\rm Cos}\nolimits} }^3}{x^2} - 3{\mathop{\rm Cos}\nolimits} {x^2}).dx} \)\(\)

=\(4\int {x*{\mathop{\rm Sin}\nolimits} {x^2}} *{\cos ^3}{x^2} - 3\int {x*{\mathop{\rm Sin}\nolimits} {x^2}*{\mathop{\rm Cos}\nolimits} {x^2}.dx} \)

Step 2 – Evaluate the integral by substituting \({\mathop{\rm Cos}\nolimits} {x^2} = t\)

Let \({\mathop{\rm Cos}\nolimits} {x^2} = t\)

Differentiating both sides with respect to x

=\( - 2x*{\mathop{\rm Sin}\nolimits} {x^2}.dx = dt\)

=\(x*{\mathop{\rm Sin}\nolimits} {x^2}.dx = ( - )\frac{{dt}}{2}\)

The integral becomes,

=\(4\int {{t^3}( - \frac{{dt}}{2}) - 3\int {t( - \frac{{dt}}{2})} } \)

=\( - 2\int {{t^3}dt + \frac{3}{2}\int {t.dt} } \)

=\( - 2*\frac{{{t^4}}}{4} + \frac{3}{2}*\frac{{{t^2}}}{2} + c\)

=\(\)\( - \frac{{{t^4}}}{2} + \frac{3}{4}*\frac{{{t^2}}}{{}} + c\)

Step 3 –(Put \(t = {\mathop{\rm Cos}\nolimits} {x^2}\)in the value of integral)=\( - \frac{1}{2}{{\mathop{\rm Cos}\nolimits} ^4}{x^2} + \frac{3}{2}{{\mathop{\rm Cos}\nolimits} ^2}{x^2} + c\)

Hence, \(\int {x*{\mathop{\rm Sin}\nolimits} {x^2}} *{\mathop{\rm Cos}\nolimits} 3{x^2}.dx\)= \( - \frac{1}{2}{{\mathop{\rm Cos}\nolimits} ^4}{x^2} + \frac{3}{2}{{\mathop{\rm Cos}\nolimits} ^2}{x^2} + c\)

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