Anti-Derivative ∫(1 + tan²x) dx
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Evaluate the indefinite integral:
![\mathsf{\displaystyle\int\!(1+tan^2\,x)\,dx}\\\\\\ \mathsf{=\displaystyle\int\!\bigg[1+\left(\frac{sin\,x}{cos\,x}\right)^{\!2}\bigg]dx}\\\\\\ \mathsf{=\displaystyle\int\!\bigg[1+\frac{sin^2\,x}{cos^2\,x}\bigg]dx}](https://tex.z-dn.net/?f=%5Cmathsf%7B%5Cdisplaystyle%5Cint%5C%21%281%2Btan%5E2%5C%2Cx%29%5C%2Cdx%7D%5C%5C%5C%5C%5C%5C%20%5Cmathsf%7B%3D%5Cdisplaystyle%5Cint%5C%21%5Cbigg%5B1%2B%5Cleft%28%5Cfrac%7Bsin%5C%2Cx%7D%7Bcos%5C%2Cx%7D%5Cright%29%5E%7B%5C%212%7D%5Cbigg%5Ddx%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7B%3D%5Cdisplaystyle%5Cint%5C%21%5Cbigg%5B1%2B%5Cfrac%7Bsin%5E2%5C%2Cx%7D%7Bcos%5E2%5C%2Cx%7D%5Cbigg%5Ddx%7D)
Reduce both terms to the same common denominator:
![\mathsf{=\displaystyle\int\!\bigg[\frac{cos^2\,x}{cos^2\,x}+\frac{sin^2\,x}{cos^2\,x}\bigg]dx}\\\\\\ \mathsf{=\displaystyle\int\!\frac{cos^2\,x+sin^2\,x}{cos^2\,x}\,dx\qquad\quad (but~~cos^2\,x+sin^2\,x=1)}\\\\\\ \mathsf{=\displaystyle\int\! \frac{1}{cos^2\,x}\,dx}\\\\\\ \mathsf{=\displaystyle\int\! sec^2\,x\,dx}](https://tex.z-dn.net/?f=%5Cmathsf%7B%3D%5Cdisplaystyle%5Cint%5C%21%5Cbigg%5B%5Cfrac%7Bcos%5E2%5C%2Cx%7D%7Bcos%5E2%5C%2Cx%7D%2B%5Cfrac%7Bsin%5E2%5C%2Cx%7D%7Bcos%5E2%5C%2Cx%7D%5Cbigg%5Ddx%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7B%3D%5Cdisplaystyle%5Cint%5C%21%5Cfrac%7Bcos%5E2%5C%2Cx%2Bsin%5E2%5C%2Cx%7D%7Bcos%5E2%5C%2Cx%7D%5C%2Cdx%5Cqquad%5Cquad%20%28but~~cos%5E2%5C%2Cx%2Bsin%5E2%5C%2Cx%3D1%29%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7B%3D%5Cdisplaystyle%5Cint%5C%21%20%5Cfrac%7B1%7D%7Bcos%5E2%5C%2Cx%7D%5C%2Cdx%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7B%3D%5Cdisplaystyle%5Cint%5C%21%20sec%5E2%5C%2Cx%5C%2Cdx%7D)
and that last one is an immediate integral:
Therefore,
✔
I hope this helps. =)
Tags: <em>indefinite integral anti-derivative trigonometric trig function tangent secant sine cosine tan sec sin cos differential integral calculus</em>
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Evaluate the indefinite integral:
Reduce both terms to the same common denominator:
and that last one is an immediate integral:
Therefore,
I hope this helps. =)
Tags: <em>indefinite integral anti-derivative trigonometric trig function tangent secant sine cosine tan sec sin cos differential integral calculus</em>
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Answer:
Step-by-step explanation:
The inequality that can be used to determine h is shown below:-
As per the given situation, it is mentioned that a plumber could earns $50 for a repair job, so for three jobs he earned $150 as a total amount
Now
We presume h be the number of hours
Also he earns additional $75 per hour worked so $75
Total amount is $150 + $75h)
The total amount is more than $600
So, the inequality equation is as given below: