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Evaluate \(\int \cos ^{3} x d x\)

(1) \(\int \cos ^{3} x d x=\int \cos ^{2} x \cdot \cos x d x\)

(2)                     \(=\int\left(1-\sin ^{2} x\right) \cos x d x\)

(3) let \(t=\sin x \longrightarrow d t=\cos x d x\)


\(\int \cos ^{3} x d x=\int\left(1-t^{2}\right) d t\)

\(\int \cos ^{3} x d x=t-\frac{t^{3}}{3}+c \)


\(\int \cos ^{3} x d x=\sin (x)-\frac{\sin ^{3} x}{3}+c \)

Evaluate \(\int \cos ^{4} x d x\)

\(\sin ^{\circ}(x)=1\)

\(0,4\)

\(\int \cos ^{4} x d x=\int\left(\cos ^{2} x\right)^{2} d x\)

                         \(=\int\left[\frac{1}{2}(1+\cos 2 x)\right]^{2} d x\)

                         \(=\frac{1}{4} \int\left[1+2 \cos 2 x+\cos ^{2} 2 x\right] d x\)

\(\cos ^{2} x=\frac{1}{2}[1+\cos 2 x]\)

\(\cos ^{2}(2 x)=\frac{1}{2}(1+\cos 4 x)\)

\(\int \cos ^{4} x d x=\frac{1}{4} \int\left[1+2 \cos 2 x+\frac{1}{2}(1+\cos 4 x)\right] d x\)

\(=\frac{1}{4} \int\left[1+2 \cos 2 x+\frac{1}{2}+\frac{1}{2} \cos 4x\right] d x\)

\(\int \cos ^{4} x d x=\frac{1}{4} \int\left(\frac{3}{2}+2 \cos 2 x+\frac{1}{2} \cos 4 x\right) d x\)

\(=\frac{1}{4}\left[\frac{3}{2} x+\sin 2 x+\frac{1}{8} \sin 4 x+c\right]\)

\(\int \cos ^{4} x d x=\frac{3}{8} x+\frac{1}{4} \sin (2 x)+\frac{1}{32} \sin (4 x)+C\)

Evaluate \(\int \sin (4 x) \cos (5 x) d x\)

\(I=\int \sin (4 x) \cos (5 x) d x\)

\(I=\int \frac{1}{2}[\sin (4 x-5 x)+\sin (4 x+5 x)] d x\)

\(=\frac{1}{2} \int[\sin (-x)+\sin (9 x)] d x\)

\(\sin (-a)=-\sin a\)

\(=\frac{1}{2} \int[-\sin x+\sin 9 x] d x\)

\(=\frac{1}{2}\left[-(-\cos x)+\frac{-1}{9} \cos 9 x\right]+c\)

\(=\frac{1}{2} \cos x-\frac{1}{18} \cos (9 x)+c\)

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