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受欢迎的微积分 >

integral of (cos(x))/((4-sin^2(x)))

解答

\int \frac{cos ( x )}{( 4 - sin ^{2} ( x ) )} d x

解答

\frac{1}{4} (ln \frac{sin ( x )}{2} + 1 - ln \frac{sin ( x )}{2} - 1) + C

求解步骤

\int \frac{cos ( x )}{4 - sin ^{2} ( x )} d x

使用换元积分法

= \int \frac{1}{4 - u ^{2}} d u

使用换元积分法

= \int \frac{1}{2 ( - v ^{2} + 1 )} d v

提出常数:

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

= \frac{1}{2} \cdot \int \frac{1}{- v ^{2} + 1} d v

使用积分常见定则:

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

= \frac{1}{2} (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2})

代回

= \frac{1}{2} \frac{ln \frac{s i n ( x )}{2} + 1}{2} - \frac{ln \frac{s i n ( x )}{2} - 1}{2}

化简

\frac{1}{2} \frac{ln \frac{s i n ( x )}{2} + 1}{2} - \frac{ln \frac{s i n ( x )}{2} - 1}{2} : \frac{1}{4} (ln \frac{sin ( x )}{2} + 1 - ln \frac{sin ( x )}{2} - 1)

= \frac{1}{4} (ln \frac{sin ( x )}{2} + 1 - ln \frac{sin ( x )}{2} - 1)

解答补常数

= \frac{1}{4} (ln \frac{sin ( x )}{2} + 1 - ln \frac{sin ( x )}{2} - 1) + C

作图

流行的例子

integral of ((8q^3-3q^2-2q)/(4q))

\int (\frac{8 q ^{3} - 3 q ^{2} - 2 q}{4 q}) d q

integral of (1-x^{-1/3})

\int (1 - x^{- \frac{1}{3}}) d x

integral of 1/((z+4)^4)

\int \frac{1}{( z + 4 ) ^{4}} d z

integral of (x^2)/(sqrt(16x^2+81))

\int \frac{x ^{2}}{16 x ^{2} + 81} d x

integral of (ln(x))/(x^{20)}

\int \frac{ln ( x )}{x ^{20}} d x