Video Transcript
Determine the integral of seven minus nine sin nine π₯ multiplied by π 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 π’ equal seven π₯ plus cos of nine π₯. This is the exponent or power of π. Differentiating this gives us seven minus nine sin of nine π₯. We differentiate cos nine π₯ by recalling that the differential of cos ππ₯ is negative π multiplied by sin ππ₯. Whilst the dπ’ and dπ₯ technically canβt be separated, for the purposes of integration by substitution, we can write that dπ’ 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 dπ’. Using our substitutions, the integration becomes the integral of π to the power of π’ dπ’. We know that the integral of π to the power of π₯ with respect to π₯ is π to the power of π₯ plus the constant π. This means that the integral of π to the power of π’ is also π to the power of π’. The final step is to substitute back the value of π’ into our answer.
The definite integral of seven minus nine sin nine π₯ multiplied by π to the power of seven π₯ plus cos nine π₯ is π to the power of seven π₯ plus cos nine π₯ plus the constant π. We could check our answer by differentiating this term, which would in turn give us the initial expression we were trying to integrate.
