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Indefinite integralsCalculusCollegeabout 0:41

Integral tan(x) dx: animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: -ln(abs(cos(x))) + C.

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Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.3 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Evaluating an integral
$\int \tan\left( x\right)\,dx$
2. Step by step
1. $\int \tan\left( x\right)\,dx$
  Start with the integral
2. $\int \tan\left( x\right)\,dx = -\ln\left(\left|\cos\left( x\right)\right|\right)$
  ∫ tan x dx = −ln|cos x|
3. $\int \tan\left( x\right)\,dx = -\ln\left(\left|\cos\left( x\right)\right|\right) + C$
  Simplify and add the constant of integration
3. The answer
$-\ln\left(\left|\cos\left( x\right)\right|\right) + C$
∫ tan(x) dx = -ln(abs(cos(x))) + C
4. Check
- Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

The problem. Here is the problem: integral tan(x) dx. Find the indefinite integral (antiderivative) of tan(x) with respect to x.

Step by step. Start with the integral. the integral of tan x dx equals negative ln|cos x|. Simplify and add the constant of integration.

The answer. So the answer is negative ln(abs(cos(x))) plus C. An antiderivative F satisfies F prime of x equals f(x). Because the derivative of any constant is 0, every antiderivative differs by a constant C.

Check. Differentiated the answer and compared it with the integrand at 8 sample points, which checks out.

About indefinite integrals

Find antiderivatives with standard rules, substitution and integration by parts.

Integral tan(x) dx: Indefinite integrals Step-by-Step Lesson