StudySquare
Integration of exponentials and logarithms for OCR A-level Maths
This page covers the following topics:
1. Integrating basic exponent
2. Integrating exponents
3. Integrating natural logarithm
4. Integrating logarithms
To find the integral of equations or variables in the form eˣ, use the following rule.
Integrate variables in the form aˣ using the exponentials rule. Integrate 2 functions multiplied together by parts; for u, choose the function that is simpler when differentiated.
Integrate variables in the form lnx using integration by parts: ∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx. Let dv/dx = 1 and u = the variable in the form lnx.
Integrate logarithmic functions using integration by parts: ∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx. Let dv/dx = 1 and u = the logarithmic function.
1
Evaluate ∫ 5ˣ ⁻ ² + 4²ˣ dx.
(5ˣ ⁻ ²)/ln5 + (2⁴ˣ ⁻ ²)/ln2
2
Find ∫ xe²ˣ dx.
Let u = x so du/dx = 1. Let dv/dx = e²ˣ so v = (1/2)e²ˣ. ∫ xe²ˣ dx = (1/2) xe²ˣ − ∫ (1/2)e²ˣ dx = (1/2) xe²ˣ − (1/4)e²ˣ + c.
3
What is ∫ 5ln(2x + 1) dx? Use the substitution a = 2x + 1.
a = 2x + 1 so dx = (1/2) da. Re−write in terms of a: 5∫ ln(a)(1/2) da. Integrate by parts and rewrite in terms of x: 5xln(2x + 1) + (5/2)ln(2x + 1) − 5x − (5/2) + c.
4
Integrate (eˣ − 2)²/e²ˣ dx.
x − 2e⁻²ˣ + 4e⁻ˣ + c.
5
Integrate ∫ 5ᵉˣ dx using the substitution u = ex.
u = ex so dx = (1/e) du. Rewrite in terms of u: ∫ (1/e) 5ᵘ du. Integrate and rewrite in terms of x: (5ᵉˣ)/eln(5) + c.
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