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1-й пример:
![5x+1 equiv t, int {5sqrt{t}} , cdot dt = \
=5int {sqrt{t}} , cdot dt = 5 cdot {frac{2sqrt{t^3}}{3}} = {frac{10sqrt{t^3}}{3}} = {frac{10sqrt{(5x+1)^3}}{3}} = \
intlimits^3_0 {sqrt{5x+1}} , dx = {frac{10sqrt{(5x+1)^3}}{3}} |^{3}_{0}= \
={frac{10sqrt{(5 cdot 3+1)^3}}{3}} - {frac{10sqrt{(5 cdot 0 +1)^3}}{3}} = frac {640}{3}-frac{10}{3}=\
=frac{630}{3}=210 5x+1 equiv t, int {5sqrt{t}} , cdot dt = \
=5int {sqrt{t}} , cdot dt = 5 cdot {frac{2sqrt{t^3}}{3}} = {frac{10sqrt{t^3}}{3}} = {frac{10sqrt{(5x+1)^3}}{3}} = \
intlimits^3_0 {sqrt{5x+1}} , dx = {frac{10sqrt{(5x+1)^3}}{3}} |^{3}_{0}= \
={frac{10sqrt{(5 cdot 3+1)^3}}{3}} - {frac{10sqrt{(5 cdot 0 +1)^3}}{3}} = frac {640}{3}-frac{10}{3}=\
=frac{630}{3}=210](https://tex.z-dn.net/?f=5x%2B1+equiv+t%2C++int+%7B5sqrt%7Bt%7D%7D+%2C+cdot+dt+%3D+%5C%0A%3D5int+%7Bsqrt%7Bt%7D%7D+%2C+cdot+dt+%3D+5+cdot+%7Bfrac%7B2sqrt%7Bt%5E3%7D%7D%7B3%7D%7D++%3D+%7Bfrac%7B10sqrt%7Bt%5E3%7D%7D%7B3%7D%7D+%3D+%7Bfrac%7B10sqrt%7B%285x%2B1%29%5E3%7D%7D%7B3%7D%7D+%3D++%5C%0A+intlimits%5E3_0+%7Bsqrt%7B5x%2B1%7D%7D+%2C+dx+%3D+%7Bfrac%7B10sqrt%7B%285x%2B1%29%5E3%7D%7D%7B3%7D%7D+%7C%5E%7B3%7D_%7B0%7D%3D++%5C%0A%3D%7Bfrac%7B10sqrt%7B%285+cdot+3%2B1%29%5E3%7D%7D%7B3%7D%7D+-+%7Bfrac%7B10sqrt%7B%285+cdot+0+%2B1%29%5E3%7D%7D%7B3%7D%7D+%3D+frac+%7B640%7D%7B3%7D-frac%7B10%7D%7B3%7D%3D%5C%0A%3Dfrac%7B630%7D%7B3%7D%3D210)
2-й пример:
![int {cos 2x} , dx = frac {sin 2x}{2} \
intlimits^{frac{ pi }{3}}_{-frac{pi}{6}} {cos 2x} , dx = frac {sin 2x}{2} |^{frac{ pi }{3}}_{-frac{pi}{6}} = \
= frac {sin frac{ 2 cdot pi }{3}}{2} - frac {sin frac{ 2 cdot pi }{6}}{2} = \
= frac {-1}{2} - frac {frac {sqrt{3}}{2}}{2} = -frac{2+sqrt{3}}{4} int {cos 2x} , dx = frac {sin 2x}{2} \
intlimits^{frac{ pi }{3}}_{-frac{pi}{6}} {cos 2x} , dx = frac {sin 2x}{2} |^{frac{ pi }{3}}_{-frac{pi}{6}} = \
= frac {sin frac{ 2 cdot pi }{3}}{2} - frac {sin frac{ 2 cdot pi }{6}}{2} = \
= frac {-1}{2} - frac {frac {sqrt{3}}{2}}{2} = -frac{2+sqrt{3}}{4}](https://tex.z-dn.net/?f=+int+%7Bcos++2x%7D+%2C+dx+%3D+frac+%7Bsin++2x%7D%7B2%7D+%5C%0A+intlimits%5E%7Bfrac%7B+pi+%7D%7B3%7D%7D_%7B-frac%7Bpi%7D%7B6%7D%7D+%7Bcos++2x%7D+%2C+dx+%3D++frac+%7Bsin++2x%7D%7B2%7D+%7C%5E%7Bfrac%7B+pi+%7D%7B3%7D%7D_%7B-frac%7Bpi%7D%7B6%7D%7D+%3D++%5C%0A%3D++frac+%7Bsin++frac%7B+2+cdot+pi+%7D%7B3%7D%7D%7B2%7D+-++frac+%7Bsin++frac%7B+2+cdot+pi+%7D%7B6%7D%7D%7B2%7D+%3D+%5C%0A%3D++frac+%7B-1%7D%7B2%7D+-++frac+%7Bfrac+%7Bsqrt%7B3%7D%7D%7B2%7D%7D%7B2%7D+%3D+-frac%7B2%2Bsqrt%7B3%7D%7D%7B4%7D)
3-й пример:
![int {x cdot e^x} , dx = e^x cdot (x-1) +C \
intlimits^1_0 {x cdot e^x} , dx = e^x cdot (x-1)|^{1}_{0} = e^1 cdot (1-1) - e^0 cdot (0-1) = \
= e cdot 0 - 1 cdot (-1) = 0 + 1 = 1 int {x cdot e^x} , dx = e^x cdot (x-1) +C \
intlimits^1_0 {x cdot e^x} , dx = e^x cdot (x-1)|^{1}_{0} = e^1 cdot (1-1) - e^0 cdot (0-1) = \
= e cdot 0 - 1 cdot (-1) = 0 + 1 = 1](https://tex.z-dn.net/?f=+int+%7Bx+cdot+e%5Ex%7D+%2C+dx+%3D+e%5Ex+cdot+%28x-1%29+%2BC+%5C%0A+intlimits%5E1_0+%7Bx+cdot+e%5Ex%7D+%2C+dx+%3D+e%5Ex+cdot+%28x-1%29%7C%5E%7B1%7D_%7B0%7D+%3D+e%5E1+cdot+%281-1%29+-+e%5E0+cdot+%280-1%29+%3D+%5C%0A%3D+e+cdot+0+-+1+cdot+%28-1%29+%3D+0+%2B+1+%3D+1)
2-й пример:
3-й пример:
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