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受欢迎的微积分 >

integral of sqrt(1+cos(6x))

解答

\int 1 + cos (6 x) d x

解答

\frac{2}{3} sin (3 x) + C

求解步骤

\int 1 + cos (6 x) d x

使用换元积分法

= \int \frac{1}{2} 1 + cos (3 u) d u

提出常数:

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

= \frac{1}{2} \cdot \int 1 + cos (3 u) d u

使用换元积分法

= \frac{1}{2} \cdot \int \frac{1}{2} 1 + cos (\frac{3 v}{2}) d v

提出常数:

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

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int 1 + cos (\frac{3 v}{2}) d v

使用换元积分法

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int 1 + cos (w) \frac{2}{3} d w

提出常数:

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

= \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{2}{3} \cdot \int 1 + cos (w) d w

使用换元积分法

= \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{2}{3} \cdot \int \frac{2 2}{( 1 + t ^{2} ) 1 + t ^{2}} d t

提出常数:

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

= \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{2}{3} \cdot 22 \cdot \int \frac{1}{( 1 + t ^{2} ) 1 + t ^{2}} d t

化简

(1 + t^{2}) 1 + t^{2} : (1 + t^{2})^{\frac{3}{2}}

= \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{2}{3} \cdot 22 \cdot \int \frac{1}{( 1 + t ^{2} ) ^{\frac{3}{2}}} d t

使用三角换元法

= \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{2}{3} \cdot 22 \cdot \int cos (r) d r

使用积分常见定则:

\int cos (r) d r = sin (r)

= \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{2}{3} \cdot 22 sin (r)

代回

= \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{2}{3} \cdot 22 sin (arctan (tan (\frac{\frac{3 \cdot 2 \cdot 2 x}{2}}{2})))

化简

\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{2}{3} \cdot 22 sin (arctan (tan (\frac{\frac{3 \cdot 2 \cdot 2 x}{2}}{2}))) : \frac{2}{3} sin (3 x)

= \frac{2}{3} sin (3 x)

解答补常数

= \frac{2}{3} sin (3 x) + C

作图

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\int \frac{4}{100 + x} d x

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\int 5 sec (x) tan (x) cos (4 + sec (x)) d x

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\int \frac{16}{x 1 + 2 x} d x