Question

Evaluate the integral from e to e^2 of ((ln^2(x)/x))dx using u-substitution. 

          Evaluate the integral from e to e^2 of ((ln^2(x)/x))dx using u-substitution.

Added by Margaret A.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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Evaluate the integral from e to e^2 of ((ln^2(x)/x))dx using u-substitution.
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Transcript

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00:01 In this question the given integral is that is integral e to the power 3x cos x d x now in the question it is saying that we have to evaluate this given integral so let's start solving this question so first i am considering the given integral that is equal to i so it will be so it will be written as that is i is equal to integral e to the power 3x cos x d x now next step is i using a formula which is used in integration by part so formula is that is integral u dv by d x multiplied by d x is equal to uv minus integral v du by d x multiplied by d x so this is a formula which is which i am using in above integral so here consider first that is u is equal to e to the power 3x and dv by d x is equal to cos x now so d u by d x will become after taking differentiation on both sides so i can say that d u by d x is equal to 3e to the power 3x and here taking indication on both side then i will get that is v is equal to sine x now next step is substitute this data in the given in the formula so it will be written as that is integral u dv by d x u is that is e to the power 3x and dv by d x is that is cose x then d x is equal to uv u is that is e to the power 3x and v that is that is sine x minus integral v that is sine x du by dx, that is 3e to the power 3x, then dx.
02:19 So i can say that, again simplify it, then this will become that is i.
02:27 So i is equal to e to the power 3x, sine x minus here.
02:34 This three will come outside.
02:39 And in the integral inside remain is, that is minus three integral.
02:44 To the power 3x, sine x d x.
02:49 Now consider it that is i1...
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