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Indefinite integralsCalculusCollegeabout 1:01

Evaluate ∫ e^x cos(x) dx: animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: e^x·(cos(x) + sin(x)) / 2 + 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 (0.4 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Evaluating an integral
$\int {e}^{x} \cos\left( x\right)\,dx$
2. Step by step
1. $\int {e}^{x} \cos\left( x\right)\,dx$
  Start with the integral
2. $\int e^{ax}\cos bx\,dx = \frac{e^{ax}(a\cos bx + b\sin bx)}{a^2+b^2}$
  Integrate by parts twice
3. $\int {e}^{x} \cos\left( x\right)\,dx = \frac{{e}^{x} \left(\cos\left( x\right)+\sin\left( x\right)\right)}{2}$
  Exponential × trigonometric (by parts twice)
4. $\int {e}^{x} \cos\left( x\right)\,dx = \frac{{e}^{x} \left(\cos\left( x\right)+\sin\left( x\right)\right)}{2} + C$
  Simplify and add the constant of integration
3. The answer
$\frac{{e}^{x} \left(\cos\left( x\right)+\sin\left( x\right)\right)}{2} + C$
∫ e^x cos(x) dx = e^x·(cos(x) + sin(x)) / 2 + C
4. Check
- Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

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

Step by step. Start with the integral. Integrate by parts twice. Integrating by parts twice returns the original integral I, so I equals e^ ax (a cos bx plus b sin bx)/(a squared plus b squared) with a equals 1, b equals 1. Exponential times trigonometric (by parts twice). Simplify and add the constant of integration.

The answer. So the answer is e to the power x times (cos(x) plus sin(x)) / 2 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.

Evaluate ∫ e^x cos(x) dx: Animated Step-by-Step Lesson