Integral of tan(x)
The integral of tan(x) is −ln(∣cos(x)∣)+C.
Graph of f(x) = tan(x) with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 0.61562647.How to work it out
- 1
We want an antiderivative of f(x) = tan(x).
∫tan(x)dx - 2
Method: standard integral.
- 3
That gives the antiderivative; add the constant C because any constant disappears when you differentiate:
∫tan(x)dx=−ln(∣cos(x)∣)+C - 4
Check by differentiating the answer; you must get the original function back:
dxd[−ln(∣cos(x)∣)]=−∣cos(x)∣sign(cos(x))−sin(x) - 5
As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and 1:
∫01tan(x)dx=[−ln(∣cos(x)∣)]01≈0.61562647
LaTeX source of the formulas on this page
\[ \int \tan\left(x\right)\,dx = -\ln\left(\left|\cos\left(x\right)\right|\right) + C \]
\[ \int \tan\left(x\right)\,dx \]
\[ \frac{d}{dx}\left[-\ln\left(\left|\cos\left(x\right)\right|\right)\right] = -\frac{\operatorname{sign}\left(\cos\left(x\right)\right) -\sin\left(x\right)}{\left|\cos\left(x\right)\right|} \]
\[ \int_{0}^{1} \tan\left(x\right)\,dx = \Big[-\ln\left(\left|\cos\left(x\right)\right|\right)\Big]_{0}^{1} \approx 0.61562647 \]Definite integrals of the same function
| Interval | Value |
|---|
| from 0 to 1 | 0.61562647 |
| from 0 to 0.5 | 0.13058424 |
Questions
What is the integral of tan(x)?
The integral of tan(x) is −ln(∣cos(x)∣)+C.
Why do we add + C?
Every function F(x) + C has the same derivative, so the antiderivative is only defined up to a constant.
What is the integral of tan(x) from 0 to 1?
About 0.61562647.