What is the integral of fe^{j^2pift} ?

The integral of fe^{j^2pift} is 1/(pi^2t^2j^4)(e^{tj^2fpi}tj^2fpi-e^{tj^2fpi})+C

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integral of fe^{j^2pift}

\int f e^{j^{2} π f t}

\frac{1}{π ^{2} t ^{2} j ^{4}} (e^{t j^{2} f π} t j^{2} f π - e^{t j^{2} f π}) + C

Solution steps

\int f e^{j^{2} π f t} df

Apply u-substitution

: \int \frac{e ^{u} u}{π ^{2} t ^{2} j ^{4}} d u

= \int \frac{e ^{u} u}{π ^{2} t ^{2} j ^{4}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{π ^{2} t ^{2} j ^{4}} \cdot \int e^{u} u d u

Apply Integration By Parts

: e^{u} u - \int e^{u} d u

= \frac{1}{π ^{2} t ^{2} j ^{4}} (e^{u} u - \int e^{u} d u)

\int e^{u} d u = e^{u}

= \frac{1}{π ^{2} t ^{2} j ^{4}} (e^{u} u - e^{u})

Substitute back

u = t j^{2} f π

= \frac{1}{π ^{2} t ^{2} j ^{4}} (e^{t j^{2} f π} t j^{2} f π - e^{t j^{2} f π})

Add a constant to the solution

= \frac{1}{π ^{2} t ^{2} j ^{4}} (e^{t j^{2} f π} t j^{2} f π - e^{t j^{2} f π}) + C

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What is the integral of fe^{j^2pift} ?
The integral of fe^{j^2pift} is 1/(pi^2t^2j^4)(e^{tj^2fpi}tj^2fpi-e^{tj^2fpi})+C