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Worked solution

Practise composite exponential forms

Integrate with respect to xx: (a) e3x−e−xe^{3x}-e^{-x}; (b) xex2xe^{x^2}; (c) etan⁡xsec⁡2xe^{\tan x}\sec^2x.

CJC · Lecture · Self-Practice 3

(a)

∫(e3x−e−x) dx=13e3x+e−x+C\int(e^{3x}-e^{-x})\,dx=\frac13e^{3x}+e^{-x}+C.

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(b)

∫xex2 dx=12∫2xex2 dx=12ex2+C\int xe^{x^2}\,dx=\frac12\int2xe^{x^2}\,dx=\frac12e^{x^2}+C.

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(c)

Use ddxtan⁡x=sec⁡2x\frac{d}{dx}\tan x=\sec^2x: ∫etan⁡xsec⁡2x dx=etan⁡x+C\int e^{\tan x}\sec^2x\,dx=e^{\tan x}+C, on each interval where cos⁡x≠0\cos x\ne0.

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Common mistakes

The minus sign before e to the negative x does not remain a minus sign after integration.

Read the hint