Which properties of integral and cos are used here?2l∫0lxcos⁡kπxl,dx=2lπ2∫0πxcos⁡kxdxwhere x from l to l.

2l∫0lxcos⁡kπxl,dx=2lπ2∫0πxcos⁡kxdxThis is a substitution, of the form u=πxlSubstitution into the first integral gives2l∫0πlπucos⁡ku⋅lu,du=2lπ2∫0lucos⁡kudu=2lπ2∫0lxcos⁡kxdxwhere the last step is just relabelling u by x since they are both dummy variables.

Answered: "Which properties of integral and cos are used..."

College
Calculus and Analysis
Integral Calculus
Applications of integrals

Babolkybya

Answered question

2022-03-13

Which properties of integral and cos are used here?

\frac{2}{l} \int_{0}^{l} x \cos \frac{k π x}{l}, d x = \frac{2 l}{π^{2}} \int_{0}^{π} x \cos k x d x

where x from l to l.

Answer & Explanation

klepbroek31s

Beginner2022-03-14Added 4 answers

\frac{2}{l} \int_{0}^{l} x \cos \frac{k π x}{l}, d x = \frac{2 l}{π^{2}} \int_{0}^{π} x \cos k x d x

This is a substitution, of the form

u = \frac{π x}{l}

Substitution into the first integral gives

\frac{2}{l} \int_{0}^{π} \frac{l}{π} u \cos k u \cdot \frac{l}{u}, d u = \frac{2 l}{π^{2}} \int_{0}^{l} u \cos k u d u = \frac{2 l}{π^{2}} \int_{0}^{l} x \cos k x d x

where the last step is just relabelling u by x since they are both dummy variables.

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