Determine the definite integral of four-sevenths sec squared seven 𝑥 over five plus one with respect to 𝑥.
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 𝑥, then d𝑦 by d𝑥, the differential, is equal to sec squared 𝑥. As integration is the inverse, or opposite, of differentiation, this implies that the integral of sec squared 𝑥 with respect to 𝑥 is equal to tan 𝑥 plus 𝑐. We can use this to solve any integral of an expression in the form 𝑛 multiplied by sec squared of 𝑎𝑥 plus 𝑏 with respect to 𝑥.
This will be equal to 𝑛 divided by 𝑎, which is the differential of 𝑎𝑥 plus 𝑏, multiplied by tan of 𝑎𝑥 plus 𝑏 plus a constant 𝑐. In this question, the value of 𝑛 is four-sevenths. 𝑎𝑥 plus 𝑏 is equal to seven-fifths 𝑥 plus one. Therefore, 𝑎 is equal to seven-fifths. To calculate 𝑛 over 𝑎, or 𝑛 divided by 𝑎, 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 𝑥 over five plus one with respect to 𝑥 is equal to 20 over 49 tan of seven 𝑥 over five plus one plus a constant 𝑐.