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Find \(\int \sec ^{2}(\sqrt{x}) d x\)

let \(t=\sqrt{x} \rightarrow x=t^{2} \rightarrow d x=2 t d t\)

\(\int \sec ^{2}(t)(2 t d t)=2 \int t \sec ^{2}(t) d t=2 J\)

\(u=t \quad \Rightarrow d u=1 d t\)

\(v=\tan (t) \Leftarrow d v=\sec ^{2}(t) d t\)

\(J=\int t \sec ^{2}(t) d(t)=\int u \cdot d v\)

\(=u \cdot v-\int v \cdot d u=t \cdot \tan (t)-\int \tan (t) \cdot d t\)

\(* \int \tan (x)=\ln |\sec (x)|\)

\(J=t \cdot \tan (t)-\ln |\sec (t)|+c\)

\(2 J=2 \cdot t+\tan (t)-2 \ln | \sec (t) |+C\)

\(I=2 \cdot \sqrt{x} \tan (\sqrt{x})-2 \ln |\sec (\sqrt{x})|+c\)

\(I=\int \sec ^{2}(\sqrt{x}) d x=2 \sqrt{x} \tan (\sqrt{x})-2 \ln |\sec \sqrt{x}|+c\)

Find \(\int x^{2} e^{x} d x\)

\(u=x^{2} \Rightarrow d u=2 x d x\)

\(v=e^{x} \Leftarrow d v=e^{x} d x\)

\(\int u d v=u \cdot v-\int v \cdot d u\)

\(\int x^{2} e^{x} d x=x^{2} \cdot e^{x}-\int e^{x} \cdot 2 x d x\)

\(=x^{2} e^{x}-2 \int x \cdot e^{x} d x=x^{2} e^{x}-2 I\)

For \(I=\int x \cdot e^{x} d x\)

\(u=x \quad \Rightarrow d u=1 d x\)

\(v=e^{x} \Leftarrow d v=e^{x} d x\)

\(I=u \cdot v - \int v d u=x e^{x}-\int e^{x} d x\)

\(I=x e^{x}-e^{x}+c\)

\(\int x^{2} e^{x} d x=x^{2} e^{x}-2 I=x^{2} e^{x}-2\left[x e^{x}-e^{x}+c\right]\)

\(=x^{2} e^{x}-2 x e^{x}+2 e^{x}-2 c\)

\(=x^{2} e^{x}-2 x e^{x}+2 e^{x}+C_{1}\)

where \(c_{1}=-2 c\)

\(=e^{x}\left(x^{2}-2 x+2\right)+c_{1}\)

Find \(\int e^{x} \cos (x) d(x)\)

\(u=\cos (x) \Rightarrow d u=-\sin (x) d x\)

\(v=e^{x} \quad \Leftarrow d v=e^{x} d x\)

\(I=\int e^{x} \cos (x) d x=u \cdot v-\int v d u\)

\(=e^{x} \cos (x)-\int -e^{x} \sin (x) d x\)

\(=e^{x} \cos (x)+\int e^{x} \sin (x) d x\)

\(=e^{x} \cos (x)+J\)

\(J=\int e^{x} \sin (x) d x\)

\(u=\sin (x) \Rightarrow d u=\cos (x) d x\)

\(v=e^{x} \Leftarrow d v=e^{x} d x\)

\(J=u \cdot v-\int v d u\)

\(J=\sin (x) e^{x}-\int e^{x} \cos x d x\)

\(J=e^{x} \sin (x)-I\)

\(J=e^{x} \sin (x)-I\)

\(I=\int e^{x} \cos (x) d x=e^{x} \cos (x)+J\)

\(I=\int e^{x} \cos (x) d x=e^{x} \cos (x)+e^{x} \sin (x)-I\)

\(2 I=e^{x} \cos (x)+e^{x} \sin (x)\)

\(\frac{2 I}{2}=\frac{e^{x} \cos (x)}{2}+\frac{x}{2} \sin (x)\)

\(\Rightarrow I=\frac{1}{2} e^{x} \cos (x)+\frac{1}{2} e^{x} \sin (x)\)

\(I=\frac{1}{2} e^{x}[\cos (x)+\sin (x)]+C\)

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