MAA9P
https://peda.net/id/993b2f3ce77
2020-08-26T11:46:14+03:00
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Tehtäviä
https://peda.net/id/e132ff9ae77
2020-08-26T11:48:14+03:00
232<br/>
<div>a)</div>
<div><img src="https://xn--lksyvihko-v2a.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D8e%5E%7B-4x%7Ddx%3D-2%5Cint_%7B%20%7D%5E%7B%20%7D-4e%5E%7B-4x%7Ddx%3D-2e%5E%7B-4x%7D%2BC" alt="\int_{ }^{ }8e^{-4x}dx=-2\int_{ }^{ }-4e^{-4x}dx=-2e^{-4x}+C"/></div>
<div>b)</div>
<div><img src="https://xn--lksyvihko-v2a.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7De%5E%7B%5Cfrac%7Bx%7D%7B3%7D%7Ddx%3D3%5Cint_%7B%20%7D%5E%7B%20%7D%5Cfrac%7B1%7D%7B3%7De%5E%7B%5Cfrac%7Bx%7D%7B3%7D%7Ddx%3D3e%5E%7B%5Cfrac%7Bx%7D%7B3%7D%7D%2BC" alt="\int_{ }^{ }e^{\frac{x}{3}}dx=3\int_{ }^{ }\frac{1}{3}e^{\frac{x}{3}}dx=3e^{\frac{x}{3}}+C"/></div>
<div>c)</div>
<div><img src="https://xn--lksyvihko-v2a.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D6xe%5E%7Bx%5E2%7Ddx%3D3%5Cint_%7B%20%7D%5E%7B%20%7D2xe%5E%7Bx%5E2%7Ddx%3D3e%5E%7Bx%5E2%7D" alt="\int_{ }^{ }6xe^{x^2}dx=3\int_{ }^{ }2xe^{x^2}dx=3e^{x^2}"/></div>
<div>233</div>
<div>a)</div>
<div><img src="https://xn--lksyvihko-v2a.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7Dt%5Csin3t%5E2dt%3D%5Cfrac%7B1%7D%7B6%7D%5Cint_%7B%20%7D%5E%7B%20%7D6t%5Csin3t%5E2dt%3D-%5Cfrac%7B1%7D%7B6%7D%5Ccos3t%5E2%2BC" alt="\int_{ }^{ }t\sin3t^2dt=\frac{1}{6}\int_{ }^{ }6t\sin3t^2dt=-\frac{1}{6}\cos3t^2+C"/></div>
<div>b)</div>
<div><img src="https://xn--lksyvihko-v2a.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D3t%5E2%5Ccos2t%5E3dt%3D%5Cfrac%7B1%7D%7B2%7D%5Cint_%7B%20%7D%5E%7B%20%7D6t%5E2%5Ccos2t%5E3dt%3D%5Cfrac%7B1%7D%7B2%7D%5Csin2t%5E3dt%2BC" alt="\int_{ }^{ }3t^2\cos2t^3dt=\frac{1}{2}\int_{ }^{ }6t^2\cos2t^3dt=\frac{1}{2}\sin2t^3dt+C"/></div>
<div>c)</div>
<div><img src="https://xn--lksyvihko-v2a.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D16t%5Csin4t%5E2dt%3D2%5Cint_%7B%20%7D%5E%7B%20%7D8t%5Csin4t%5E2dt%3D-2%5Ccos4t%5E2%2BC" alt="\int_{ }^{ }16t\sin4t^2dt=2\int_{ }^{ }8t\sin4t^2dt=-2\cos4t^2+C"/><br/>
234<br/>
<div>integroidaan funktio </div>
<div><img src="https://xn--lksyvihko-v2a.fi/math.svg?latex=f%5Cleft%28x%5Cright%29%3D2%5Csin%20x%2B%5Ccos2x" alt="f\left(x\right)=2\sin x+\cos2x"/><br/>
<img src="https://xn--lksyvihko-v2a.fi/math.svg?latex=F%5Cleft(x%5Cright)%3D%5Cint_%7B%20%7D%5E%7B%20%7D2%5Csin%20x%2B%5Ccos%20xdx%3D-2%5Ccos%20x%2B%5Cfrac%7B1%7D%7B2%7D%5Csin%20x%2BC" alt="F\left(x\right)=\int_{ }^{ }2\sin x+\cos xdx=-2\cos x+\frac{1}{2}\sin x+C"/><br/>
<img src="https://xn--lksyvihko-v2a.fi/math.svg?latex=F%5Cleft%28%5Cpi%5Cright%29%3D3" alt="F\left(\pi\right)=3"/><br/>
<img src="https://xn--lksyvihko-v2a.fi/math.svg?latex=F%5Cleft%28%5Cpi%5Cright%29%3D-2%5Ccos%5Cpi%2B%5Cfrac%7B1%7D%7B2%7D%5Csin%5Cpi%2BC%3D-2%5Ccdot%5Cleft%28-1%5Cright%29%2B%5Cfrac%7B1%7D%7B2%7D%5Ccdot0%2BC%3D2%2BC" alt="F\left(\pi\right)=-2\cos\pi+\frac{1}{2}\sin\pi+C=-2\cdot\left(-1\right)+\frac{1}{2}\cdot0+C=2+C"/><br/>
<img src="https://xn--lksyvihko-v2a.fi/math.svg?latex=2%2BC%3D3%7B%2C%7D%5C%20C%3D1" alt="2+C=3{,}\ C=1"/></div>
<div>Haluttu funktio on <img src="https://xn--lksyvihko-v2a.fi/math.svg?latex=-2%5Ccos%20x%2B%5Cfrac%7B1%7D%7B2%7D%5Csin%20x%2B1" alt="-2\cos x+\frac{1}{2}\sin x+1"/><br/>
<br/>
</div>
240<br/>
<div>a) <img src="https://xn--lksyvihko-v2a.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Csin%20xdx%3D-%5Ccos%20x%2BC" alt="\int_{ }^{ }\sin xdx=-\cos x+C"/></div>
<div>D</div>
<div>b)</div>
<div><img src="https://xn--lksyvihko-v2a.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Ccos%20xdx%3D%5Csin%20x%2BC" alt="\int_{ }^{ }\cos xdx=\sin x+C"/></div>
<div>B</div>
<div>c)</div>
<div><img src="https://xn--lksyvihko-v2a.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7De%5Exdx%3De%5Ex%2BC" alt="\int_{ }^{ }e^xdx=e^x+C"/></div>
<div>C<br/>
242<br/>
<br/>
</div>
<div>a)</div>
<div><img src="https://xn--lksyvihko-v2a.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft%28e%5Ex%2B1%5Cright%29%5E2dx%3D%5Cint_%7B%20%7D%5E%7B%20%7De%5E%7B2x%7D%2B2e%5Ex%2B1dx%3D%5Cfrac%7B1%7D%7B2%7De%5E%7B2x%7D%2B2e%5Ex%2Bx%2BC" alt="\int_{ }^{ }\left(e^x+1\right)^2dx=\int_{ }^{ }e^{2x}+2e^x+1dx=\frac{1}{2}e^{2x}+2e^x+x+C"/></div>
<div>b)</div>
<div><img src="https://xn--lksyvihko-v2a.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7De%5Ex%5Cleft%28e%5Ex%2B1%5Cright%29%5E3dx%3D%5Cfrac%7B1%7D%7B4%7D%5Cleft%28e%5Ex%2B1%5Cright%29%5E4%2BC" alt="\int_{ }^{ }e^x\left(e^x+1\right)^3dx=\frac{1}{4}\left(e^x+1\right)^4+C"/></div>
<div>c)</div>
<div><img src="https://xn--lksyvihko-v2a.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Csin%20x%5Ccos%5E2xdx%3D-%5Cint_%7B%20%7D%5E%7B%20%7D-%5Csin%20x%5Ccos%5E2xdx%3D-%5Cfrac%7B1%7D%7B3%7D%5Ccos%5E2x%2BC" alt="\int_{ }^{ }\sin x\cos^2xdx=-\int_{ }^{ }-\sin x\cos^2xdx=-\frac{1}{3}\cos^2x+C"/></div>
</div>
2020-08-26T11:48:14+03:00