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2.2 Übungen

Aus Online Mathematik Brückenkurs 2

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Version vom 13:05, 16. Sep. 2008

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Exercise 2.2:1

Calculate the integrals

a)	\displaystyle \displaystyle \int_{1}^{2} \displaystyle\frac{dx}{(3x-1)^4}\quad by using the substitution \displaystyle u=3x-1,
b)	\displaystyle \displaystyle \int (x^2+3)^5x \, dx\quad by using the substitution \displaystyle u=x^2+3,
c)	\displaystyle \displaystyle \int x^2 e^{x^3} \, dx\quad by using the substitution \displaystyle u=x^3.

Answer

Solution a

Solution b

Solution c

Exercise 2.2:2

Calculate the integrals

a)	\displaystyle \displaystyle\int_{0}^{\pi} \cos 5x\, dx	b)	\displaystyle \displaystyle\int_{0}^{1/2} e^{2x+3}\, dx
c)	\displaystyle \displaystyle\int_{0}^{5} \sqrt{3x + 1} \, dx	d)	\displaystyle \displaystyle\int_{0}^{1} \sqrt[\scriptstyle3]{1 - x}\, dx

Answer

Solution a

Solution b

Solution c

Solution d

Exercise 2.2:3

Calculate the integrals

a)	\displaystyle \displaystyle\int 2x \sin x^2\, dx	b)	\displaystyle \displaystyle\int \sin x \cos x\, dx
c)	\displaystyle \displaystyle\int \displaystyle\frac{\ln x}{x}\, dx	d)	\displaystyle \displaystyle\int \displaystyle\frac{x+1}{x^2+2x+2}\, dx
e)	\displaystyle \displaystyle\int \displaystyle\frac{3x}{x^2+1}\, dx	f)	\displaystyle \displaystyle\int \displaystyle\frac{\sin \sqrt{x}}{\sqrt{x}}\, dx

Answer

Solution a

Solution b

Solution c

Solution d

Solution e

Solution f

Exercise 2.2:4

Use the formula

\displaystyle \int \frac{dx}{x^2+1} = \arctan x + C

to calculate the integrals

a)	\displaystyle \displaystyle\int \frac{dx}{x^2+4}	b)	\displaystyle \displaystyle\int \frac{dx}{(x-1)^2+3}
c)	\displaystyle \displaystyle\int \frac{dx}{x^2+4x+8}	d)	\displaystyle \displaystyle\int \frac{x^2}{x^2 +1}\, dx