Video Transcript
Determine the integral of seven minus nine sin nine 𝑥 multiplied by 𝑒 to the power of seven 𝑥 plus cos nine 𝑥 with respect to 𝑥.
We will solve this integration by substitution. And we begin by letting 𝑢 equal seven 𝑥 plus cos of nine 𝑥. This is the exponent or power of 𝑒. Differentiating this gives us seven minus nine sin of nine 𝑥. We differentiate cos nine 𝑥 by recalling that the differential of cos 𝑛𝑥 is negative 𝑛 multiplied by sin 𝑛𝑥. Whilst the d𝑢 and d𝑥 technically can’t be separated, for the purposes of integration by substitution, we can write that d𝑢 is equal to seven minus nine sin of nine 𝑥 d𝑥.
We notice that these are the other two parts of the initial integration. We can replace these two parts with d𝑢. Using our substitutions, the integration becomes the integral of 𝑒 to the power of 𝑢 d𝑢. We know that the integral of 𝑒 to the power of 𝑥 with respect to 𝑥 is 𝑒 to the power of 𝑥 plus the constant 𝑐. This means that the integral of 𝑒 to the power of 𝑢 is also 𝑒 to the power of 𝑢. The final step is to substitute back the value of 𝑢 into our answer.
The definite integral of seven minus nine sin nine 𝑥 multiplied by 𝑒 to the power of seven 𝑥 plus cos nine 𝑥 is 𝑒 to the power of seven 𝑥 plus cos nine 𝑥 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.
