CalculusEasyMade_CH15

15_1

$\int \sqrt{a^{2} - x^{2}}\;dx$?
이 함수의 적분을 위해 부분적분과 치환적분을 적용합니다. (예 5 참조)
$u=\sqrt{a^{2} - x^{2} , \qard du=dx$

a, x, u=symbols('a x u', positive=True)
y=sqrt(a**2-x**2)
y
$\sqrt{a^{2} - x^{2}}$
u1=y
du=diff(x, x)
du
1

r1=u1*integrate(du, x)-diff(u1, x)*integrate(du, x)
r1
$\frac{x^{2}}{\sqrt{a^{2} - x^{2}}} + x \sqrt{a^{2} - x^{2}}$

위 마지막 결과의 첫번 째항을 적분하기 위해 다음과 같이 정리하여 실시합니다.
$\begin{align}\frac{x^{2}}{\sqrt{a^{2} - x^{2}}}&=-\frac{-a^2+a^2-x^{2}}{\sqrt{a^{2} - x^{2}}}\\&=\frac{a^2}{\sqrt{a^{2} - x^{2}}}-\frac{a^2-x^{2}}{\sqrt{a^{2} - x^{2}}}\\ &=\frac{a^2}{\sqrt{a^{2} - x^{2}}}-{\sqrt{a^{2} - x^{2}}}\end{align}$
위 최종 정리된 항의 첫 번째식의 적분은 $x=asin(\theta)$로 치환하여 실시하면 다음과 같습니다.

integrate(a**2/sqrt(a**2-x**2), x)
$a^{2} \operatorname{asin}{\left(\frac{x}{a} \right)}$

또한 두번째 항은 원래의 적분항과 같습니다. 이를 고려하면 최종 결과는 다음과 같습니다.

(integrate(a**2/sqrt(a**2-x**2), x)+x*sqrt(a**2-x**2))/2
$\frac{a^{2} \operatorname{asin}{\left(\frac{x}{a} \right)}}{2} + \frac{x \sqrt{a^{2} - x^{2}}}{2}$
integrate(y, x)

$\frac{a^{2} \operatorname{asin}{\left(\frac{x}{a} \right)}}{2} + \frac{x \sqrt{a^{2} - x^{2}}}{2}$

15_2

x=symbols('x', positive=True)
y=x*log(x)
y
xlog(x)
u1=log(x)
dv1=x*diff(x, x)
r1=u1*integrate(dv1, x)-integrate(diff(u1, x)*integrate(dv1, x), x)
r1
$\frac{x^{2} \log{\left(x \right)}}{2} - \frac{x^{2}}{4}$

integrate(x*log(x), x)
$\frac{x^{2} \log{\left(x \right)}}{2} - \frac{x^{2}}{4}$

15_3

$\int x^alog_e(x)\;dx$ ?

a, x=symbols('a x', positive=True)
y=x**a*log(x)
y
xalog(x)

u=log(x)
dv=x**a*diff(x, x)
dv
xa

r=u1*integrate(dv, x)-integrate(diff(u, x)*integrate(dv, x), x)

r
$- \frac{x x^{a}}{\left(a + 1\right)^{2}} + \frac{x^{a + 1} \log{\left(x \right)}}{a + 1}$

factor(r)
$\frac{x x^{a} \left(a \log{\left(x \right)} + \log{\left(x \right)} - 1\right)}{\left(a + 1\right)^{2}}$

factor(integrate(y, x))
$\frac{x x^{a} \left(a \log{\left(x \right)} + \log{\left(x \right)} - 1\right)}{\left(a + 1\right)^{2}}$

15_4

$\int e^xcos(e^x) dx$ ?
$u=e^x, du=e^x dx \rightarrow \int cos(u) du$

a, x=symbols('a x', positive=True)
y=exp(x)*cos(exp(x))
y
$e^{x} \cos{\left(e^{x} \right)}$

u1=exp(x)
du=diff(u1, x)
du
e^x

u=symbols('u', real=True)
y1=cos(u)
int_y1=integrate(y1, u)
int_y1
sin(u)

int_y1.subs(u, u1)
sin(e^x)

int_y=integrate(exp(x)*cos(exp(x)), x)
int_y
sin(e^x)

15_5

a, x=symbols('a x', positive=True)
y=1/x*cos(log(x))
y
$\frac{cos(log(x))}{x}$
u1=log(x)
du=diff(u1, x)
du
$\frac{1}{x}$
u=symbols('u', real=True)
y1=cos(u)
int_y1=integrate(y1, u)
int_y1
sin(u)
int_y1.subs(u, u1)
sin(log(x))
int_y=integrate(y, x)
int_y
sin(log(x))

15_6

$\begin{align}\int x^2e^xdx &=x^2\int e^xdx-\int \left((x^2)\prime \int e^x dx \right)\\ &= x^2e^x-2\int xe^x dx\end{align}$
위 결과의 마지막 항에 다시 부분적분을 적용합니다.
$\begin{align}\int xe^xdx &=x\int e^xdx-\int \left((x)\prime \int e^x dx \right)\\ &= xe^x-\int e^x dx\\ &= xe^x- e^x\end{align}$
위 두 결과를 합하면 다음과 같습니다.
$\int x^2e^xdx =x^2e^x-2(xe^x- e^x)$

a, x=symbols('a x', positive=True)
y=x**2*exp(x)
y
x²e^x
integrate(y, x)
(x²−2x+2)e^x

15_7

a, x=symbols('a x', positive=True)
y=(log(x)**a/x)
y
\frac{\log{\left(x \right)}^{a}}{x}

u1=log(x)
du1=diff(log(x))
du1
$\frac{1}{x}$

u=symbols('u', real=True)
y1=u**a
int_y1=integrate(y1, u)
int_y1
$\frac{u^{a + 1}}{a + 1}$

int_y1.subs(u, u1)
$\frac{\log{\left(x \right)}^{a + 1}}{a + 1}$

integrate(y, x)
$\frac{\log{\left(x \right)}^{a + 1}}{a + 1}$

15_8

a, x=symbols('a x', positive=True)
y=1/(log(x)*x)
y
$\frac{1}{xlog(x)}$

u1=log(x)
du1=diff(log(x))
du1
$\frac{1}{x}$

u=symbols('u', real=True)
y1=1/u
int_y1=integrate(y1, u)
int_y1
log(u)

int_y1.subs(u, u1)
log(log(x))

integrate(y, x)
log(log(x))

15_9

a, x=symbols('a x', positive=True)
y=(5*x+1)/(x**2+x-2)
y
$\frac{5 x + 1}{x^{2} + x - 2}$

y1=apart(y)
y1
$$\frac{3}{x+2}+\frac{2}{x-1}$

integrate(3/(x+2), x)+integrate(2/(x-1), x)
2log(x−1)+3log(x+2)

integrate(y, x)
2log(x−1)+3log(x+2)

15_10

a, x=symbols('a x', positive=True)
y=(x**2-3)/(x**3-7*x+6)
y
$\frac{x^{2} - 3}{x^{3} - 7 x + 6}$

y1=apart(y, full=True)
y1
$\frac{3}{10 \left(x + 3\right)} + \frac{1}{2 \left(x - 1\right)} + \frac{1}{5 \left(x - 2\right)}$

integrate(3/(10*(x+2)), x)+integrate(1/(2*(x-1)), x)+integrate(1/(5*(x-2)), x)
$\frac{\log{\left(2 x - 2 \right)}}{2} + \frac{\log{\left(5 x - 10 \right)}}{5} + \frac{3 \log{\left(10 x + 20 \right)}}{10}$

integrate(y, x)
$\frac{\log{\left(2 x - 2 \right)}}{2} + \frac{\log{\left(5 x - 10 \right)}}{5} + \frac{3 \log{\left(10 x + 20 \right)}}{10}$

15_11

a, b, x=symbols('a b x', positive=True)
y=(b/(x**2-a**2))
y
$\frac{b}{- a^{2} + x^{2}}$

y1=apart(y, x)
y1
$- \frac{b}{2 a \left(a + x\right)} + \frac{b}{2 a \left(- a + x\right)}$

integrate(-b/(2*a*(a+x)), x)+integrate(b/(2*a*(x-a)), x)
$ \frac{b \log{\left(- 2 a^{2} + 2 a x \right)}}{2 a} - \frac{b \log{\left(2 a^{2} + 2 a x \right)}}{2 a}$

integrate(y, x)
$b \left(\frac{\log{\left(- a + x \right)}}{2 a} - \frac{\log{\left(a + x \right)}}{2 a}\right)$

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