Video: Finding the Integration of a Function Involving Trigonometric and Exponential Functions Using Integration by Substitution

Determine ∫(7 βˆ’ 9 sin 9π‘₯) 𝑒^(7π‘₯ + cos 9π‘₯) dπ‘₯.

02:12

Video Transcript

Determine the integral of seven minus nine sin nine π‘₯ multiplied by 𝑒 to the power of seven π‘₯ plus cos nine π‘₯ with respect to π‘₯.

We will solve this integration by substitution. And we begin by letting 𝑒 equal seven π‘₯ plus cos of nine π‘₯. This is the exponent or power of 𝑒. Differentiating this gives us seven minus nine sin of nine π‘₯. We differentiate cos nine π‘₯ by recalling that the differential of cos 𝑛π‘₯ is negative 𝑛 multiplied by sin 𝑛π‘₯. Whilst the d𝑒 and dπ‘₯ technically can’t be separated, for the purposes of integration by substitution, we can write that d𝑒 is equal to seven minus nine sin of nine π‘₯ dπ‘₯.

We notice that these are the other two parts of the initial integration. We can replace these two parts with d𝑒. Using our substitutions, the integration becomes the integral of 𝑒 to the power of 𝑒 d𝑒. We know that the integral of 𝑒 to the power of π‘₯ with respect to π‘₯ is 𝑒 to the power of π‘₯ plus the constant 𝑐. This means that the integral of 𝑒 to the power of 𝑒 is also 𝑒 to the power of 𝑒. The final step is to substitute back the value of 𝑒 into our answer.

The definite integral of seven minus nine sin nine π‘₯ multiplied by 𝑒 to the power of seven π‘₯ plus cos nine π‘₯ is 𝑒 to the power of seven π‘₯ plus cos nine π‘₯ plus the constant 𝑐. We could check our answer by differentiating this term, which would in turn give us the initial expression we were trying to integrate.

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