Parçalı (kısmi) integrale ilişkin zor problemler

Problem 1

integral, e, start superscript, x, end superscript, sine, x, space, d, x, equals
1 cevap seçin:
1 cevap seçin:

Bunu, kısmi integral almayı iki kez kullanarak çözeceğiz.
integral, u, space, d, v, equals, u, v, minus, integral, v, space, d, u, space olduğunu hatırlayın.
Bu problemde, space, u, equals, e, start superscript, x, end superscript, space ve space, d, v, equals, sine, left parenthesis, x, right parenthesis, space, d, x, space diyeceğiz. Bu durumda, space, d, u, equals, e, start superscript, x, end superscript, space, d, x, space ve space, v, equals, integral, sine, x, space, d, x, equals, minus, cosine, x, space.
Parçalı integral bunu verir:
space, integral, e, start superscript, x, end superscript, sine, x, space, d, x, equals, minus, e, start superscript, x, end superscript, cosine, x, plus, integral, e, start superscript, x, end superscript, cosine, x, space, d, x, space
integral, e, start superscript, x, end superscript, cosine, x, space, d, x, space'i bulmak için gene parçalı integral alacağız. Bu kez space, u, equals, e, start superscript, x, end superscript, space ve space, d, v, equals, cosine, x, space, d, x, space diyeceğiz. Bu durumda, space, d, u, equals, e, start superscript, x, end superscript, space, d, x, space ve space, v, equals, sine, x, space'tir.
Şimdi şunu elde ederiz:
space, integral, e, start superscript, x, end superscript, sine, x, space, d, x, equals, minus, e, start superscript, x, end superscript, cosine, x, plus, e, start superscript, x, end superscript, sine, x, minus, integral, e, start superscript, x, end superscript, sine, x, space, d, x, space
Denklemdeki benzer terimlere dikkat edin:
space, start color blueD, integral, e, start superscript, x, end superscript, sine, x, space, d, x, end color blueD, equals, minus, e, start superscript, x, end superscript, cosine, x, plus, e, start superscript, x, end superscript, sine, x, start color blueD, minus, integral, e, start superscript, x, end superscript, sine, x, space, d, x, space, end color blueD.
İki tarafa da integral, e, start superscript, x, end superscript, sine, x, space, d, x, space, space eklediğimizde bunu elde ediyoruz:
space, start color blueD, 2, integral, e, start superscript, x, end superscript, sine, x, space, d, x, end color blueD, equals, minus, e, start superscript, x, end superscript, cosine, x, plus, e, start superscript, x, end superscript, sine, x, plus, C, space
Her iki tarafı 2 ile bölme bunu verir:
space, integral, e, start superscript, x, end superscript, sine, x, space, d, x, equals, start fraction, 1, divided by, 2, end fraction, e, start superscript, x, end superscript, left parenthesis, sine, x, minus, cosine, x, right parenthesis, plus, C, space

Problem 2

integral, left parenthesis, natural log, x, right parenthesis, start superscript, 2, end superscript, space, d, x, space, equals
1 cevap seçin:
1 cevap seçin:

Bunu, kısmi integral almayı iki kez kullanarak çözeceğiz.
integral, u, space, d, v, equals, u, v, minus, integral, v, space, d, u, space olduğunu hatırlayın.
Bu problemde, space, u, equals, left parenthesis, natural log, x, right parenthesis, start superscript, 2, end superscript, space ve space, d, v, equals, d, x, space diyeceğiz. Bu durumda, space, d, u, equals, 2, natural log, x, dot, start fraction, 1, divided by, x, end fraction, space, d, x, space ve space, v, equals, x, space.
Parçalı integral bunu verir:
space, integral, left parenthesis, natural log, x, right parenthesis, start superscript, 2, end superscript, d, x, equals, x, left parenthesis, natural log, x, right parenthesis, start superscript, 2, end superscript, minus, 2, integral, natural log, x, space, d, x, space
integral, natural log, x, space, d, x, space'i bulmak için gene parçalı integral alacağız. Bu kez space, u, equals, natural log, x, space ve space, d, v, equals, d, x, space diyeceğiz. Bu durumda, space, d, u, equals, start fraction, 1, divided by, x, end fraction, space, d, x, space ve space, v, equals, x, space'tir.
Şimdi şunu elde ederiz:
(lnx)2dx=x(lnx)22(xlnxx1xdx)=x(lnx)22xlnx+21dx=x(lnx)22xlnx+2x+C\begin{aligned}\displaystyle\qquad\int (\ln x)^2dx &= x(\ln x)^2-2\left(x\ln x-\int x\cdot\dfrac1x\,dx\right)\\\\\\\\ &=x(\ln x)^2-2x\ln x+2\int 1\, dx\\\\\\\\ &=x(\ln x )^2-2x\ln x +2x+C\end{aligned}

Problem 3

integral, x, start superscript, 2, end superscript, sine, left parenthesis, pi, x, right parenthesis, space, d, x, space, equals
1 cevap seçin:
1 cevap seçin:

Bunu, kısmi integral almayı iki kez kullanarak çözeceğiz.
integral, u, space, d, v, equals, u, v, minus, integral, v, space, d, u, space olduğunu hatırlayın.
Bu problemde, space, u, equals, x, start superscript, 2, end superscript, space ve space, d, v, equals, sine, left parenthesis, pi, x, right parenthesis, space, d, x, space diyeceğiz. Bu durumda, space, d, u, equals, 2, x, space, d, x, space ve space, v, equals, integral, sine, left parenthesis, pi, x, right parenthesis, space, d, x, equals, minus, start fraction, cosine, left parenthesis, pi, x, right parenthesis, divided by, pi, end fraction, space.
Parçalı integral bunu verir:
=x2sin(πx) dx=x2cos(πx)π2xcos(πx)πdx=x2cos(πx)π+2πxcos(πx)dx\begin{aligned} &\phantom{=}\displaystyle\int x^2\sin(\pi x)\ dx \\\\ &= -\dfrac{x^2\cos(\pi x)}{\pi}-\int-\dfrac{2x\cos(\pi x)}{\pi}\, dx \\\\ &= -\dfrac{x^2\cos(\pi x)}{\pi}+\dfrac2\pi\int{x\cos(\pi x)}\, dx\, \end{aligned}
integral, x, cosine, left parenthesis, pi, x, right parenthesis, space, d, x, space'i bulmak için gene parçalı integral alacağız. Bu kez space, u, equals, x, space ve space, d, v, equals, cosine, left parenthesis, pi, x, right parenthesis, space, d, x, space diyeceğiz. Bu durumda, space, d, u, equals, d, x, space ve space, v, equals, integral, cosine, left parenthesis, pi, x, right parenthesis, space, d, x, equals, start fraction, sine, left parenthesis, pi, x, right parenthesis, divided by, pi, end fraction, space'dir.
Şimdi şunu elde ederiz:
=x2sin(πx) dx=x2cos(πx)π+2π(xsin(πx)πsin(πx)πdx)=x2cos(πx)π+2xsin(πx)π22π2sin(πx)dx=x2cos(πx)π+2xsin(πx)π22π21π(cos(πx))=x2cos(πx)π+2xsin(πx)π2+2cos(πx)π3+C\begin{aligned} &\phantom{=}\displaystyle\int x^2\sin(\pi x)\ dx \\\\ &= -\dfrac{x^2\cos(\pi x)}{\pi}+\dfrac2\pi\left(\dfrac{x\sin(\pi x)}{\pi}-\int\dfrac{\sin(\pi x)}{\pi}\,dx\right) \\\\ &= -\dfrac{x^2\cos(\pi x)}{\pi}+\dfrac{2x\sin(\pi x)}{\pi^2}-\dfrac{2}{\pi^2}\int{\sin(\pi x)}\,dx \\\\ &=-\dfrac{x^2\cos(\pi x)}{\pi}+\dfrac{2x\sin(\pi x)}{\pi^2}-\dfrac{2}{\pi^2}\cdot\dfrac{1}{\pi}(-\cos(\pi x)) \\\\ &=-\dfrac{x^2\cos(\pi x)}{\pi}+\dfrac{2x\sin(\pi x)}{\pi^2}+\dfrac{2\cos(\pi x)}{\pi^3}+C \end{aligned}