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受欢迎的微积分 >

integral of (1+cos^2(5)x^2csc^2(5)x^2)6x

解答

\int (1 + cos^{2} (5) x^{2} csc^{2} (5) x^{2}) 6 x d x

解答

6 csc^{2} (5) (\frac{sin ^{2} ( 5 ) x ^{2}}{2} + \frac{cos ^{2} ( 5 ) x ^{6}}{6}) + C

求解步骤

\int (1 + cos^{2} (5) x^{2} csc^{2} (5) x^{2}) \cdot 6 x d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 6 \cdot \int (1 + cos^{2} (5) x^{2} csc^{2} (5) x^{2}) x d x

使用三角恒等式改写

= 6 \cdot \int \frac{x ( sin ^{2} ( 5 ) + cos ^{2} ( 5 ) x ^{4} )}{sin ^{2} ( 5 )} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 6 \cdot \frac{1}{sin ^{2} ( 5 )} \cdot \int x (sin^{2} (5) + cos^{2} (5) x^{4}) d x

乘开

x (sin^{2} (5) + cos^{2} (5) x^{4}) : sin^{2} (5) x + cos^{2} (5) x^{5}

= 6 \cdot \frac{1}{sin ^{2} ( 5 )} \cdot \int sin^{2} (5) x + cos^{2} (5) x^{5} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 6 \cdot \frac{1}{sin ^{2} ( 5 )} (\int sin^{2} (5) x d x + \int cos^{2} (5) x^{5} d x)

\int sin^{2} (5) x d x = \frac{sin ^{2} ( 5 ) x ^{2}}{2}

\int cos^{2} (5) x^{5} d x = \frac{cos ^{2} ( 5 ) x ^{6}}{6}

= 6 \cdot \frac{1}{sin ^{2} ( 5 )} (\frac{sin ^{2} ( 5 ) x ^{2}}{2} + \frac{cos ^{2} ( 5 ) x ^{6}}{6})

化简

6 \cdot \frac{1}{sin ^{2} ( 5 )} (\frac{sin ^{2} ( 5 ) x ^{2}}{2} + \frac{cos ^{2} ( 5 ) x ^{6}}{6}) : 6 csc^{2} (5) (\frac{sin ^{2} ( 5 ) x ^{2}}{2} + \frac{cos ^{2} ( 5 ) x ^{6}}{6})

= 6 csc^{2} (5) (\frac{sin ^{2} ( 5 ) x ^{2}}{2} + \frac{cos ^{2} ( 5 ) x ^{6}}{6})

解答补常数

= 6 csc^{2} (5) (\frac{sin ^{2} ( 5 ) x ^{2}}{2} + \frac{cos ^{2} ( 5 ) x ^{6}}{6}) + C

作图

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