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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 eหฃ, use the following rule.

Integrating basic exponent

Integrate variables in the form aหฃ 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: โˆซ u (dv/dx) dx = uv โˆ’ โˆซ 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: โˆซ u (dv/dx) dx = uv โˆ’ โˆซ v (du/dx) dx. Let dv/dx = 1 and u = the logarithmic function.

Integrating logarithms

1

Evaluate โˆซ 5หฃ โป ยฒ + 4ยฒหฃ dx.

(5หฃ โป ยฒ)/ln5 + (2โดหฃ โป ยฒ)/ln2

Evaluate โˆซ 5หฃ โป ยฒ + 4ยฒหฃ dx.

2

Find โˆซ xeยฒหฃ dx.

Let u = x so du/dx = 1. Let dv/dx = eยฒหฃ so v = (1/2)eยฒหฃ. โˆซ xeยฒหฃ dx = (1/2) xeยฒหฃ โˆ’ โˆซ (1/2)eยฒหฃ dx = (1/2) xeยฒหฃ โˆ’ (1/4)eยฒหฃ + c.

Find โˆซ xeยฒหฃ dx.

3

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

a = 2x + 1 so dx = (1/2) da. Reโˆ’write in terms of a: 5โˆซ ln(a)(1/2) da. Integrate by parts and rewrite in terms of x: 5xln(2x + 1) + (5/2)ln(2x + 1) โˆ’ 5x โˆ’ (5/2) + c.

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

4

Integrate (eหฃ โˆ’ 2)ยฒ/eยฒหฃ dx.

x โˆ’ 2eโปยฒหฃ + 4eโปหฃ + c.

Integrate (eหฃ โˆ’ 2)ยฒ/eยฒหฃ dx.

5

Integrate โˆซ 5แต‰หฃ dx using the substitution u = ex.

u = ex so dx = (1/e) du. Rewrite in terms of u: โˆซ (1/e) 5แต˜ du. Integrate and rewrite in terms of x: (5แต‰หฃ)/eln(5) + c.

Integrate โˆซ 5แต‰หฃ dx using the substitution u = ex.

End of page

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