Medium

Integrating Exponential and Trigonometric Functions

Simple Explanation

Beyond polynomials, the exponential and trigonometric functions have their own standard antiderivatives: ∫eˣdx=eˣ+C, ∫sin(x)dx=−cos(x)+C, and ∫cos(x)dx=sin(x)+C — each the reverse of a familiar derivative rule.

Why Do We Need It?

Exponential and trigonometric functions model an enormous range of real phenomena — growth, decay, and periodic motion — so integrating them is essential for finding totals and accumulated change in those settings.

Formula

Integrals of Exponential and Trigonometric Functions

∫eˣ dx = eˣ + C; ∫sin(x) dx = −cos(x) + C; ∫cos(x) dx = sin(x) + C

The standard antiderivatives for the exponential and core trigonometric functions, each the reverse of a standard derivative rule.

C: — the constant of integration

When to use it: Whenever integrating a term that is exactly eˣ, sin(x), or cos(x) (or a constant multiple of one).

Worked Example

Integrate a mix of exponential and trigonometric terms

Find ∫(2eˣ + 3sin(x)) dx.

Why Does This Work?

Differentiating 2eˣ−3cos(x)+C gives 2eˣ−3(−sin(x)) = 2eˣ+3sin(x), which is exactly the original integrand — confirming the reverse relationship between this integral and the corresponding derivative rules.

Real-Life Example

Total sound energy from a vibrating string

The instantaneous amplitude of a vibrating string is naturally modeled with sine and cosine functions of time.

Integrating that amplitude function over an interval recovers the accumulated quantity — exactly the kind of computation ∫sin(x)dx and ∫cos(x)dx make possible.

Practice

Find ∫4cos(x) dx.

Medium

Common mistake

Mixing up the signs — ∫sin(x)dx = −cos(x)+C (with a minus sign), while ∫cos(x)dx = sin(x)+C (with no minus sign); it is easy to swap these by mistake.

Quick Review

∫eˣ dx = eˣ + C.
∫sin(x) dx = −cos(x) + C.
∫cos(x) dx = sin(x) + C.