Question

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Answer 1

$a, \int \sin^{2} x . \cos x d x = \int \sin^{2} x . d (\sin x) = \frac{1}{3} \sin^{3} x + C b, \int \cos^{3} x . \sin x d x = - \int \cos^{3} x . d (\cos x) = - \frac{1}{4} \cos^{4} x + C c, \int \cos^{3} x d x = \int \cos^{2} x . \cos x d x = \int (1 - \sin^{2} x) . \cos x d x = \int \cos x d x - \int \sin^{2} x . \cos x d x = \sin x + C - \int \sin^{2} x . d (\sin x) = \sin x - \frac{1}{3} \sin^{3} x + C$

$d, \int x . e^{1 + x^{2}} d x = \frac{1}{2} . \int e^{1 + x^{2}} . 2 x d x = \frac{1}{2} \int e^{1 + x^{2}} d (1 + x^{2}) = \frac{1}{2} e^{1 + x^{2}} + C e, \int \sin^{3} x d x = \int \sin^{2} x . \sin x d x = \int (1 - \cos^{2} x) \sin x d x = \int \sin x d x - \int \cos^{2} x \sin x d x = - \cos x + C + \int \cos^{2} x d (\cos x) = - \cos x + \frac{1}{3} \cos^{3} x + C f, \int \sin x . e^{1 + \cos x} d x = - \int e^{1 + \cos x} d (1 + \cos x) = - e^{1 + \cos x} + C$ ..

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Answer 2

$a, \int \sin^{2} x . \cos x d x = \int \sin^{2} x . d (\sin x) = \frac{1}{3} \sin^{3} x + C b, \int \cos^{3} x . \sin x d x = - \int \cos^{3} x . d (\cos x) = - \frac{1}{4} \cos^{4} x + C c, \int \cos^{3} x d x = \int \cos^{2} x . \cos x d x = \int (1 - \sin^{2} x) . \cos x d x = \int \cos x d x - \int \sin^{2} x . \cos x d x = \sin x + C - \int \sin^{2} x . d (\sin x) = \sin x - \frac{1}{3} \sin^{3} x + C$

$d, \int x . e^{1 + x^{2}} d x = \frac{1}{2} . \int e^{1 + x^{2}} . 2 x d x = \frac{1}{2} \int e^{1 + x^{2}} d (1 + x^{2}) = \frac{1}{2} e^{1 + x^{2}} + C e, \int \sin^{3} x d x = \int \sin^{2} x . \sin x d x = \int (1 - \cos^{2} x) \sin x d x = \int \sin x d x - \int \cos^{2} x \sin x d x = - \cos x + C + \int \cos^{2} x d (\cos x) = - \cos x + \frac{1}{3} \cos^{3} x + C f, \int \sin x . e^{1 + \cos x} d x = - \int e^{1 + \cos x} d (1 + \cos x) = - e^{1 + \cos x} + C$ ..

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