Integral of tan(t) dt

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$$\int\limits_{0}^{1} \tan{\left(t \right)}\, dt$$

Integral(tan(t), (t, 0, 1))

Detail solution

Rewrite the integrand:

$\tan{\left(t \right)} = \frac{\sin{\left(t \right)}}{\cos{\left(t \right)}}$
Let $u = \cos{\left(t \right)}$ .

Then let $du = - \sin{\left(t \right)} dt$ and substitute $- du$ :

$\int \left(- \frac{1}{u}\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{u}\, du = - \int \frac{1}{u}\, du$
  1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
  So, the result is: $- \log{\left(u \right)}$
Now substitute $u$ back in:

$- \log{\left(\cos{\left(t \right)} \right)}$
Add the constant of integration:

$- \log{\left(\cos{\left(t \right)} \right)}+ \mathrm{constant}$

The answer is:

$- \log{\left(\cos{\left(t \right)} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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 | tan(t) dt = C - log(cos(t))
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$$\int \tan{\left(t \right)}\, dt = C - \log{\left(\cos{\left(t \right)} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-log(cos(1))

$$- \log{\left(\cos{\left(1 \right)} \right)}$$

-log(cos(1))

$$- \log{\left(\cos{\left(1 \right)} \right)}$$

-log(cos(1))

Numerical answer [src]

0.615626470386014

0.615626470386014

The graph

Use the examples entering the upper and lower limits of integration.