Examples in integration

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Integration

In the end, most times, you can add an additive constant "+C". It is not done here.

\(\int x^5dx=\frac{x^6}{6}\)
\(\int \sin(5x)dx=-\frac15\cos(5x)\)
\(\int \cos(5x)dx=\frac15\sin(5x)\)
\(\int \exp(5x)dx=\frac15\exp(5x)\)
\(\int \ln(5x)dx=\int \ln(5)+\ln(x)dx=x\ln(5)+(x\ln(x)-x)\)
\(\int \ln(x)\cdot xdx=\ln(x)\cdot\frac{x^2}{2}-\int \frac{1}{x}\cdot\frac{x^2}{2}dx=\frac{x^2}{2}-\frac{x^2}{4}=\frac{x^2}{4}\)
\(\int x\cdot\exp(x)dx=x\cdot\exp(x)-\int 1\cdot\exp(x)dx=x\exp(x)-\exp(x)\)
\(\int x\cdot\sin(x)dx=x\cdot(-\cos(x))-\int 1\cdot(-\cos(x))dx=-x\cos(x)+\sin(x)\)
\(\int \frac{x^3}{(x-2)(x-3)}dx=\int x+5+\frac{-8}{x-2}+\frac{27}{x-3}dx=\frac{x^2}{2}+5x-8\ln|x-2|+27\ln|x-3|\)