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  • Evaluate the definite integrals in Exercises 1 to 20. 7

Evaluate the definite integrals in Exercises 1 to 20.

    Q7.    \int^\frac{\pi}{4}_0 \tan x dx

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Given integral: \int^\frac{\pi}{4}_0 \tan x dx

Consider the integral \int \tan x dx

\int \tan x dx = -\log|\cos x |

So, we have the function of x, f(x) = -\log|\cos x |

Now, by Second fundamental theorem of calculus, we have

I = f(\frac{\pi}{4}) -f(0)

= -\log\left | \cos \frac{\pi}{4} \right | +\log|\cos 0|

= -\log\left | \cos \frac{1}{\sqrt2} \right | +\log|1|

= -\log\left | \frac{1}{\sqrt2} \right | + 0 = -\log (2)^{-\frac{1}{2}}

= \frac{1}{2}\log (2)

Posted by

Divya Prakash Singh

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