Integration of exponentials and logarithms for OCR A-level Maths

Integration of exponentials and logarithms

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.

Integrating basic exponent

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.

Integrating exponents

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.

Integrating natural logarithm

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.

Integrating logarithms

1

Evaluate ∫ 5ˣ ⁻ ² + 4²ˣ dx.

(5ˣ ⁻ ²)/ln5 + (2⁴ˣ ⁻ ²)/ln2

Evaluate ∫ 5ˣ ⁻ ² + 4²ˣ dx.

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.

Find ∫ xe²ˣ dx.

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.

What is ∫ 5ln(2x + 1) dx? Use the substitution a = 2x + 1.

4

Integrate (eˣ − 2)²/e²ˣ dx.

x − 2e⁻²ˣ + 4e⁻ˣ + c.

Integrate (eˣ − 2)²/e²ˣ dx.

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.

Integrate ∫ 5ᵉˣ dx using the substitution u = ex.

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