Finding Indefinite Integrals

Find:

1. \(\int x^{4}\, dx\)
2. \(\int \sqrt{x} dx\)
3. \(\int \dfrac{\sin x}{\cos ^{2}x} dx\)

Solution (a) All the antiderivatives of \(f(x)=x^{4} \) are \(F(x) =\dfrac{x^{5}}{5}+C\), so \[ \int x^{4}\, dx=\dfrac{x^{5}}{5}+C \]

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(b) All the antiderivatives of \(f(x) =\sqrt{x} =x^{1/2} \) are \(F(x) =\dfrac{x^{3/2}}{\dfrac{3}{2}}+C=\dfrac{2x^{3/2}}{3}+C\), so \[ \int \sqrt{x} dx=\dfrac{2x^{3/2}}{3}+C \]

(c) No antiderivative in Table 1 corresponds to \(f(x)=\dfrac{\sin x}{\cos ^{2}x}\), so we begin by using trigonometric identities to rewrite \(\dfrac{\sin x}{\cos ^{2}x}\) in a form whose antiderivative is recognizable. \[ \dfrac{\sin x}{\cos ^{2}x}=\dfrac{\sin x}{\cos x\cdot \cos x}=\dfrac{1}{\cos x}\cdot \dfrac{\sin x}{\cos x}=\sec x\tan x \]

Then \[ \int \dfrac{\sin x}{\cos ^{2}x}\, dx=\int \sec x\tan x dx=\sec x+C \]