MAA9P

Tehtäviä

232
a)
\int_{ }^{ }8e^{-4x}dx=-2\int_{ }^{ }-4e^{-4x}dx=-2e^{-4x}+C
b)
\int_{ }^{ }e^{\frac{x}{3}}dx=3\int_{ }^{ }\frac{1}{3}e^{\frac{x}{3}}dx=3e^{\frac{x}{3}}+C
c)
\int_{ }^{ }6xe^{x^2}dx=3\int_{ }^{ }2xe^{x^2}dx=3e^{x^2}
233
a)
\int_{ }^{ }t\sin3t^2dt=\frac{1}{6}\int_{ }^{ }6t\sin3t^2dt=-\frac{1}{6}\cos3t^2+C
b)
\int_{ }^{ }3t^2\cos2t^3dt=\frac{1}{2}\int_{ }^{ }6t^2\cos2t^3dt=\frac{1}{2}\sin2t^3dt+C
c)
\int_{ }^{ }16t\sin4t^2dt=2\int_{ }^{ }8t\sin4t^2dt=-2\cos4t^2+C
234
integroidaan funktio
f\left(x\right)=2\sin x+\cos2x
F\left(x\right)=\int_{ }^{ }2\sin x+\cos xdx=-2\cos x+\frac{1}{2}\sin x+C
F\left(\pi\right)=3
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
2+C=3{,}\ C=1
Haluttu funktio on -2\cos x+\frac{1}{2}\sin x+1

240
a) \int_{ }^{ }\sin xdx=-\cos x+C
D
b)
\int_{ }^{ }\cos xdx=\sin x+C
B
c)
\int_{ }^{ }e^xdx=e^x+C
C
242

a)
\int_{ }^{ }\left(e^x+1\right)^2dx=\int_{ }^{ }e^{2x}+2e^x+1dx=\frac{1}{2}e^{2x}+2e^x+x+C
b)
\int_{ }^{ }e^x\left(e^x+1\right)^3dx=\frac{1}{4}\left(e^x+1\right)^4+C
c)
\int_{ }^{ }\sin x\cos^2xdx=-\int_{ }^{ }-\sin x\cos^2xdx=-\frac{1}{3}\cos^2x+C