integral of (6x^5+x^{1/4}-3)/(-5x^{3/2)}

해법

\int \frac{6 x ^{5} + x ^{\frac{1}{4}} - 3}{- 5 x ^{\frac{3}{2}}} d x

- \frac{4 x ^{5} - 12 x ^{\frac{1}{4}} + 18}{15 x ^{\frac{1}{2}}} + C

솔루션 단계

\int \frac{6 x ^{5} + x ^{\frac{1}{4}} - 3}{- 5 x ^{\frac{3}{2}}} d x

정수를 빼라:

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

= \frac{1}{- 5} \cdot \int \frac{6 x ^{5} + x ^{\frac{1}{4}} - 3}{x ^{\frac{3}{2}}} d x

단순화

= (- \frac{1}{5}) \cdot \int \frac{6 x ^{5} + x ^{\frac{1}{4}} - 3}{x ^{\frac{3}{2}}} d x

\frac{6 x ^{5} + x ^{\frac{1}{4}} - 3}{x ^{\frac{3}{2}}}

확대한다:

6 x^{\frac{7}{2}} + \frac{1}{x ^{\frac{5}{4}}} - \frac{3}{x ^{\frac{3}{2}}}

= (- \frac{1}{5}) \cdot \int 6 x^{\frac{7}{2}} + \frac{1}{x ^{\frac{5}{4}}} - \frac{3}{x ^{\frac{3}{2}}} d x

합계 규칙 적용:

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

= (- \frac{1}{5}) (\int 6 x^{\frac{7}{2}} d x + \int \frac{1}{x ^{\frac{5}{4}}} d x - \int \frac{3}{x ^{\frac{3}{2}}} d x)

\int 6 x^{\frac{7}{2}} d x = \frac{4}{3} x^{\frac{9}{2}}

\int \frac{1}{x ^{\frac{5}{4}}} d x = - \frac{4}{x ^{\frac{1}{4}}}

\int \frac{3}{x ^{\frac{3}{2}}} d x = - \frac{6}{x ^{\frac{1}{2}}}

= (- \frac{1}{5}) (\frac{4}{3} x^{\frac{9}{2}} - \frac{4}{x ^{\frac{1}{4}}} - (- \frac{6}{x ^{\frac{1}{2}}}))

(- \frac{1}{5}) (\frac{4}{3} x^{\frac{9}{2}} - \frac{4}{x ^{\frac{1}{4}}} - (- \frac{6}{x ^{\frac{1}{2}}}))

간소화하다 :

- \frac{4 x ^{5} - 12 x ^{\frac{1}{4}} + 18}{15 x ^{\frac{1}{2}}}

= - \frac{4 x ^{5} - 12 x ^{\frac{1}{4}} + 18}{15 x ^{\frac{1}{2}}}

솔루션에 상수 추가

= - \frac{4 x ^{5} - 12 x ^{\frac{1}{4}} + 18}{15 x ^{\frac{1}{2}}} + C