Lesson 9 of 13 · 7 min
Exponential pairs and square-root forms
NCERT §7.6.1–7.6.2
∫eˣ(sin x + cos x) dx looks like two rounds of integration by parts. It takes one line, because cos x is the derivative of sin x.
The lesson in notes
In short
If the integrand is eˣ times a function plus its own derivative, the answer is immediate: ∫eˣ[f(x) + f′(x)] dx = eˣ f(x) + C.
Why: by parts, ∫eˣ f dx = eˣ f − ∫eˣ f′ dx, and the leftover term cancels the ∫eˣ f′ dx already present.
Worked example: sin x and cos x are a function and its derivative, so ∫eˣ(sin x + cos x) dx = eˣ sin x + C.
Worked example: ∫eˣ(1/x − 1/x²) dx = eˣ/x + C, since the derivative of 1/x is −1/x².
∫√(x² − a²) dx = (x/2)√(x² − a²) − (a²/2) log|x + √(x² − a²)| + C.
∫√(x² + a²) dx = (x/2)√(x² + a²) + (a²/2) log|x + √(x² + a²)| + C.
∫√(a² − x²) dx = (x/2)√(a² − x²) + (a²/2) sin⁻¹(x/a) + C.
A quadratic under the root is completed to a square first. Example: x² + 2x + 5 = (x + 1)² + 2², so ∫√(x² + 2x + 5) dx = ((x + 1)/2)√(x² + 2x + 5) + 2 log|x + 1 + √(x² + 2x + 5)| + C.