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(1)
![intlimits {9x^2+5cos(x)} , dx = 9*intlimits {x^2} , dx +5* intlimits {cos(x)} , dx =\\
=9*frac{x^3}{3}+5*sin(x)+C=3x^3+5sin(x)+C intlimits {9x^2+5cos(x)} , dx = 9*intlimits {x^2} , dx +5* intlimits {cos(x)} , dx =\\
=9*frac{x^3}{3}+5*sin(x)+C=3x^3+5sin(x)+C](https://tex.z-dn.net/?f=+intlimits+%7B9x%5E2%2B5cos%28x%29%7D+%2C+dx+%3D+9%2Aintlimits+%7Bx%5E2%7D+%2C+dx+%2B5%2A+intlimits+%7Bcos%28x%29%7D+%2C+dx+%3D%5C%5C%0A%3D9%2Afrac%7Bx%5E3%7D%7B3%7D%2B5%2Asin%28x%29%2BC%3D3x%5E3%2B5sin%28x%29%2BC)
(2)
![intlimits {frac{5x^4-6}{x}} , dx = intlimits {5x^3-frac{6}{x}} , dx =\\
=5* intlimits {x^3} , dx -6* intlimits^a_b {frac{1}{x}} , dx =\\
=5*frac{x^4}{4}-6*ln|x|+C=\\
=frac{5}{4}x^4-6*ln|x|+C intlimits {frac{5x^4-6}{x}} , dx = intlimits {5x^3-frac{6}{x}} , dx =\\
=5* intlimits {x^3} , dx -6* intlimits^a_b {frac{1}{x}} , dx =\\
=5*frac{x^4}{4}-6*ln|x|+C=\\
=frac{5}{4}x^4-6*ln|x|+C](https://tex.z-dn.net/?f=+intlimits+%7Bfrac%7B5x%5E4-6%7D%7Bx%7D%7D+%2C+dx+%3D+intlimits+%7B5x%5E3-frac%7B6%7D%7Bx%7D%7D+%2C+dx+%3D%5C%5C%0A%3D5%2A+intlimits+%7Bx%5E3%7D+%2C+dx+-6%2A+intlimits%5Ea_b+%7Bfrac%7B1%7D%7Bx%7D%7D+%2C+dx+%3D%5C%5C%0A%3D5%2Afrac%7Bx%5E4%7D%7B4%7D-6%2Aln%7Cx%7C%2BC%3D%5C%5C%0A%3Dfrac%7B5%7D%7B4%7Dx%5E4-6%2Aln%7Cx%7C%2BC)
(3)
![intlimits {frac{x}{sqrt{3-2x^2}}} , dx = intlimits {frac{2x}{2sqrt{3-2x^2}}} , dx = intlimits {frac{1}{2sqrt{3-2x^2}}} , d(x^2) =\\=
intlimits {frac{1}{-4sqrt{3-2x^2}}} , d(-2x^2)=intlimits {frac{1}{-4sqrt{3-2x^2}}} , d(3-2x^2)=\\
=[t=3-2x^2]=-frac{1}{4}* intlimits {frac{1}{sqrt{t}}} , dt=-frac{1}{4}* intlimits {t^{-frac{1}{2}}} , dt=\\
=-frac{1}{4}*frac{t^{frac{1}{2}}}{frac{1}{2}}+C=-frac{sqrt{t}}{2}+C=-frac{sqrt{3-2x^2}}{2}+C intlimits {frac{x}{sqrt{3-2x^2}}} , dx = intlimits {frac{2x}{2sqrt{3-2x^2}}} , dx = intlimits {frac{1}{2sqrt{3-2x^2}}} , d(x^2) =\\=
intlimits {frac{1}{-4sqrt{3-2x^2}}} , d(-2x^2)=intlimits {frac{1}{-4sqrt{3-2x^2}}} , d(3-2x^2)=\\
=[t=3-2x^2]=-frac{1}{4}* intlimits {frac{1}{sqrt{t}}} , dt=-frac{1}{4}* intlimits {t^{-frac{1}{2}}} , dt=\\
=-frac{1}{4}*frac{t^{frac{1}{2}}}{frac{1}{2}}+C=-frac{sqrt{t}}{2}+C=-frac{sqrt{3-2x^2}}{2}+C](https://tex.z-dn.net/?f=+intlimits+%7Bfrac%7Bx%7D%7Bsqrt%7B3-2x%5E2%7D%7D%7D+%2C+dx+%3D+intlimits+%7Bfrac%7B2x%7D%7B2sqrt%7B3-2x%5E2%7D%7D%7D+%2C+dx+%3D+intlimits+%7Bfrac%7B1%7D%7B2sqrt%7B3-2x%5E2%7D%7D%7D+%2C+d%28x%5E2%29+%3D%5C%5C%3D%0Aintlimits+%7Bfrac%7B1%7D%7B-4sqrt%7B3-2x%5E2%7D%7D%7D+%2C+d%28-2x%5E2%29%3Dintlimits+%7Bfrac%7B1%7D%7B-4sqrt%7B3-2x%5E2%7D%7D%7D+%2C+d%283-2x%5E2%29%3D%5C%5C%0A%3D%5Bt%3D3-2x%5E2%5D%3D-frac%7B1%7D%7B4%7D%2A+intlimits+%7Bfrac%7B1%7D%7Bsqrt%7Bt%7D%7D%7D+%2C+dt%3D-frac%7B1%7D%7B4%7D%2A+intlimits+%7Bt%5E%7B-frac%7B1%7D%7B2%7D%7D%7D+%2C+dt%3D%5C%5C%0A%3D-frac%7B1%7D%7B4%7D%2Afrac%7Bt%5E%7Bfrac%7B1%7D%7B2%7D%7D%7D%7Bfrac%7B1%7D%7B2%7D%7D%2BC%3D-frac%7Bsqrt%7Bt%7D%7D%7B2%7D%2BC%3D-frac%7Bsqrt%7B3-2x%5E2%7D%7D%7B2%7D%2BC)
(4)
![intlimits {frac{(xsqrt{x}-3)^2}{x^3}} , dx = intlimits {frac{x^3-6x^{frac{3}{2}}-9}{x^3}} , dx = intlimits {1} , dx -6 intlimits^a_b {x^{-frac{3}{2}}} , dx - 9*intlimits {x^{-3}} , dx =\\
=x-6*frac{x^{-frac{1}{2}}}{-frac{1}{2}}-9*frac{x^{-2}}{-2}+C=\\
=x+frac{12}{sqrt{x}}+frac{9}{2x^2}+C intlimits {frac{(xsqrt{x}-3)^2}{x^3}} , dx = intlimits {frac{x^3-6x^{frac{3}{2}}-9}{x^3}} , dx = intlimits {1} , dx -6 intlimits^a_b {x^{-frac{3}{2}}} , dx - 9*intlimits {x^{-3}} , dx =\\
=x-6*frac{x^{-frac{1}{2}}}{-frac{1}{2}}-9*frac{x^{-2}}{-2}+C=\\
=x+frac{12}{sqrt{x}}+frac{9}{2x^2}+C](https://tex.z-dn.net/?f=+intlimits+%7Bfrac%7B%28xsqrt%7Bx%7D-3%29%5E2%7D%7Bx%5E3%7D%7D+%2C+dx+%3D+intlimits+%7Bfrac%7Bx%5E3-6x%5E%7Bfrac%7B3%7D%7B2%7D%7D-9%7D%7Bx%5E3%7D%7D+%2C+dx+%3D+intlimits+%7B1%7D+%2C+dx+-6+intlimits%5Ea_b+%7Bx%5E%7B-frac%7B3%7D%7B2%7D%7D%7D+%2C+dx+-+9%2Aintlimits+%7Bx%5E%7B-3%7D%7D+%2C+dx+%3D%5C%5C%0A%3Dx-6%2Afrac%7Bx%5E%7B-frac%7B1%7D%7B2%7D%7D%7D%7B-frac%7B1%7D%7B2%7D%7D-9%2Afrac%7Bx%5E%7B-2%7D%7D%7B-2%7D%2BC%3D%5C%5C%0A%3Dx%2Bfrac%7B12%7D%7Bsqrt%7Bx%7D%7D%2Bfrac%7B9%7D%7B2x%5E2%7D%2BC)
(5)
![intlimits {x^2(1+2x)} , dx = intlimits {x^2+2x^3} , dx = intlimits {x^2} , dx +2* intlimits {x^3} , dx =\\=frac{x^3}{3}+2*frac{x^4}{4}+C=
frac{x^3}{3}+frac{x^4}{2}+C intlimits {x^2(1+2x)} , dx = intlimits {x^2+2x^3} , dx = intlimits {x^2} , dx +2* intlimits {x^3} , dx =\\=frac{x^3}{3}+2*frac{x^4}{4}+C=
frac{x^3}{3}+frac{x^4}{2}+C](https://tex.z-dn.net/?f=+intlimits+%7Bx%5E2%281%2B2x%29%7D+%2C+dx+%3D+intlimits+%7Bx%5E2%2B2x%5E3%7D+%2C+dx+%3D+intlimits+%7Bx%5E2%7D+%2C+dx+%2B2%2A+intlimits+%7Bx%5E3%7D+%2C+dx+%3D%5C%5C%3Dfrac%7Bx%5E3%7D%7B3%7D%2B2%2Afrac%7Bx%5E4%7D%7B4%7D%2BC%3D%0Afrac%7Bx%5E3%7D%7B3%7D%2Bfrac%7Bx%5E4%7D%7B2%7D%2BC)
(6)
![intlimits {e^{-3x}} , dx = intlimits {frac{e^{-3x}}{-3}} , d(-3x)
=[t=-3x]=\\=-frac{1}{3}* intlimits {e^t} , dt=-frac{1}{3}e^t+C=\\
=-frac{1}{3}e^{-3x}+C intlimits {e^{-3x}} , dx = intlimits {frac{e^{-3x}}{-3}} , d(-3x)
=[t=-3x]=\\=-frac{1}{3}* intlimits {e^t} , dt=-frac{1}{3}e^t+C=\\
=-frac{1}{3}e^{-3x}+C](https://tex.z-dn.net/?f=+intlimits+%7Be%5E%7B-3x%7D%7D+%2C+dx+%3D+intlimits+%7Bfrac%7Be%5E%7B-3x%7D%7D%7B-3%7D%7D+%2C+d%28-3x%29+%0A%3D%5Bt%3D-3x%5D%3D%5C%5C%3D-frac%7B1%7D%7B3%7D%2A+intlimits+%7Be%5Et%7D+%2C+dt%3D-frac%7B1%7D%7B3%7De%5Et%2BC%3D%5C%5C%0A%3D-frac%7B1%7D%7B3%7De%5E%7B-3x%7D%2BC)
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