Question Video: Integrating Reciprocal Trigonometric Functions Mathematics • Higher Education

Determine ∫ (4/7) secΒ² ((7π‘₯/5) + 1) dπ‘₯.

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Video Transcript

Determine the definite integral of four-sevenths sec squared seven π‘₯ over five plus one with respect to π‘₯.

Initially, this might seem like quite a complicated integration problem. However, when we recall one of our standard differentials, the problem becomes much easier. We know that if 𝑦 is equal to tan π‘₯, then d𝑦 by dπ‘₯, the differential, is equal to sec squared π‘₯. As integration is the inverse, or opposite, of differentiation, this implies that the integral of sec squared π‘₯ with respect to π‘₯ is equal to tan π‘₯ plus 𝑐. We can use this to solve any integral of an expression in the form 𝑛 multiplied by sec squared of π‘Žπ‘₯ plus 𝑏 with respect to π‘₯.

This will be equal to 𝑛 divided by π‘Ž, which is the differential of π‘Žπ‘₯ plus 𝑏, multiplied by tan of π‘Žπ‘₯ plus 𝑏 plus a constant 𝑐. In this question, the value of 𝑛 is four-sevenths. π‘Žπ‘₯ plus 𝑏 is equal to seven-fifths π‘₯ plus one. Therefore, π‘Ž is equal to seven-fifths. To calculate 𝑛 over π‘Ž, or 𝑛 divided by π‘Ž, we must divide four-sevenths by seven-fifths.

Dividing by a fraction is the same as multiplying by the reciprocal of the fraction. This means that four-sevenths divided by seven-fifths is the same as four-sevenths multiplied by five-sevenths. Multiplying the numerators gives us 20. And multiplying the denominators gives us 49. We can, therefore, say that the definite integral of four-sevenths sec squared of seven π‘₯ over five plus one with respect to π‘₯ is equal to 20 over 49 tan of seven π‘₯ over five plus one plus a constant 𝑐.

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