Tabel cu integrale nedeterminate ale funcțiilor exponențiale

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Această pagină conține un tabel de integrale nedeterminate ale funcțiilor exponențiale . Pentru alte integrale, consultați Tabelele integrale .

\int e^{cx}\;\mathrm {d} x={\frac {1}{c}}e^{cx}

{\ displaystyle \ int e ^ {cx} \; \ mathrm {d} x = {\ frac {1} {c}} e ^ {cx}}

{\ displaystyle \ int e ^ {cx} \; \ mathrm {d} x = {\ frac {1} {c}} e ^ {cx}}

\int a^{cx}\;\mathrm {d} x={\frac {1}{c\log a}}a^{cx}\qquad {\mbox{(per }}a>0,{\mbox{ }}a\neq 1{\mbox{)}}

{\ displaystyle \ int a ^ {cx} \; \ mathrm {d} x = {\ frac {1} {c \ log a}} a ^ {cx} \ qquad {\ mbox {(per}} a> 0 , {\ mbox {}} a \ neq 1 {\ mbox {)}}}

{\ displaystyle \ int a ^ {cx} \; \ mathrm {d} x = {\ frac {1} {c \ log a}} a ^ {cx} \ qquad {\ mbox {(per}} a> 0 , {\ mbox {}} a \ neq 1 {\ mbox {)}}}

\int x^{n}e^{cx}\;\mathrm {d} x={\frac {1}{c}}x^{n}e^{cx}-{\frac {n}{c}}\int x^{n-1}e^{cx}\mathrm {d} x

{\ displaystyle \ int x ^ {n} e ^ {cx} \; \ mathrm {d} x = {\ frac {1} {c}} x ^ {n} e ^ {cx} - {\ frac {n } {c}} \ int x ^ {n-1} e ^ {cx} \ mathrm {d} x}

{\ displaystyle \ int x ^ {n} e ^ {cx} \; \ mathrm {d} x = {\ frac {1} {c}} x ^ {n} e ^ {cx} - {\ frac {n } {c}} \ int x ^ {n-1} e ^ {cx} \ mathrm {d} x}

care are, ca cazuri particulare:

$\int xe^{cx}\;\mathrm {d} x={\frac {e^{cx}}{c^{2}}}(cx-1)$ ${\ displaystyle \ int xe ^ {cx} \; \ mathrm {d} x = {\ frac {e ^ {cx}} {c ^ {2}}} (cx-1)}$ ${\ displaystyle \ int xe ^ {cx} \; \ mathrm {d} x = {\ frac {e ^ {cx}} {c ^ {2}}} (cx-1)}$
$\int x^{2}e^{cx}\;\mathrm {d} x=e^{cx}\left({\frac {x^{2}}{c}}-{\frac {2x}{c^{2}}}+{\frac {2}{c^{3}}}\right)$ ${\ displaystyle \ int x ^ {2} e ^ {cx} \; \ mathrm {d} x = e ^ {cx} \ left ({\ frac {x ^ {2}} {c}} - {\ frac {2x} {c ^ {2}}} + {\ frac {2} {c ^ {3}}} \ right)}$ ${\ displaystyle \ int x ^ {2} e ^ {cx} \; \ mathrm {d} x = e ^ {cx} \ left ({\ frac {x ^ {2}} {c}} - {\ frac {2x} {c ^ {2}}} + {\ frac {2} {c ^ {3}}} \ right)}$

Demonstrație

Aplic integrarea pe părți : $\int f'g\;\mathrm {d} x=fg-\int fg'\;\mathrm {d} x$ ${\ displaystyle \ int f'g \; \ mathrm {d} x = fg- \ int fg '\; \ mathrm {d} x}$ ${\ displaystyle \ int f'g \; \ mathrm {d} x = fg- \ int fg '\; \ mathrm {d} x}$
cu

$f'=e^{cx}\,\!$ ${\ displaystyle f '= e ^ {cx} \, \!}$ ${\ displaystyle f '= e ^ {cx} \, \!}$
$g=x^{n}$ ${\ displaystyle g = x ^ {n}}$ ${\ displaystyle g = x ^ {n}}$

Si in consecinta

$f={\frac {1}{c}}e^{cx}$ ${\ displaystyle f = {\ frac {1} {c}} e ^ {cx}}$ ${\ displaystyle f = {\ frac {1} {c}} e ^ {cx}}$
$g'=nx^{n-1}$ ${\ displaystyle g '= nx ^ {n-1}}$ ${\ displaystyle g '= nx ^ {n-1}}$

Vom avea atunci

\int x^{n}e^{cx}\;\mathrm {d} x={\frac {1}{c}}e^{cx}x^{n}-\int {\frac {1}{c}}e^{cx}nx^{n-1}\mathrm {d} x\;={\frac {1}{c}}x^{n}e^{cx}-{\frac {n}{c}}\int x^{n-1}e^{cx}\mathrm {d} x

{\ displaystyle \ int x ^ {n} e ^ {cx} \; \ mathrm {d} x = {\ frac {1} {c}} e ^ {cx} x ^ {n} - \ int {\ frac {1} {c}} e ^ {cx} nx ^ {n-1} \ mathrm {d} x \; = {\ frac {1} {c}} x ^ {n} e ^ {cx} - { \ frac {n} {c}} \ int x ^ {n-1} e ^ {cx} \ mathrm {d} x}

{\ displaystyle \ int x ^ {n} e ^ {cx} \; \ mathrm {d} x = {\ frac {1} {c}} e ^ {cx} x ^ {n} - \ int {\ frac {1} {c}} e ^ {cx} nx ^ {n-1} \ mathrm {d} x \; = {\ frac {1} {c}} x ^ {n} e ^ {cx} - { \ frac {n} {c}} \ int x ^ {n-1} e ^ {cx} \ mathrm {d} x}

cvd

\int {\frac {e^{cx}\;\mathrm {d} x}{x}}=\log |x|+\sum _{i=1}^{\infty }{\frac {(cx)^{i}}{i\cdot i!}}

{\ displaystyle \ int {\ frac {e ^ {cx} \; \ mathrm {d} x} {x}} = \ log | x | + \ sum _ {i = 1} ^ {\ infty} {\ frac {(cx) ^ {i}} {i \ cdot i!}}}

{\ displaystyle \ int {\ frac {e ^ {cx} \; \ mathrm {d} x} {x}} = \ log | x | + \ sum _ {i = 1} ^ {\ infty} {\ frac {(cx) ^ {i}} {i \ cdot i!}}}

\int {\frac {e^{cx}\;\mathrm {d} x}{x^{n}}}={\frac {1}{n-1}}\left(-{\frac {e^{cx}}{x^{n-1}}}+c\int {\frac {e^{cx}\mathrm {d} x}{x^{n-1}}}\right)\qquad {\mbox{(per }}n\neq 1{\mbox{)}}

{\ displaystyle \ int {\ frac {e ^ {cx} \; \ mathrm {d} x} {x ^ {n}}} = {\ frac {1} {n-1}} \ left (- {\ frac {e ^ {cx}} {x ^ {n-1}}} + c \ int {\ frac {e ^ {cx} \ mathrm {d} x} {x ^ {n-1}}} \ right ) \ qquad {\ mbox {(for}} n \ neq 1 {\ mbox {)}}}

{\ displaystyle \ int {\ frac {e ^ {cx} \; \ mathrm {d} x} {x ^ {n}}} = {\ frac {1} {n-1}} \ left (- {\ frac {e ^ {cx}} {x ^ {n-1}}} + c \ int {\ frac {e ^ {cx} \ mathrm {d} x} {x ^ {n-1}}} \ right ) \ qquad {\ mbox {(for}} n \ neq 1 {\ mbox {)}}}

\int e^{cx}\ln x\;\mathrm {d} x={\frac {1}{c}}\left(e^{cx}\log |x|-\int {\frac {e^{cx}\mathrm {d} x}{x}}\right)

{\ displaystyle \ int e ^ {cx} \ ln x \; \ mathrm {d} x = {\ frac {1} {c}} \ left (e ^ {cx} \ log | x | - \ int {\ frac {e ^ {cx} \ mathrm {d} x} {x}} \ right)}

{\ displaystyle \ int e ^ {cx} \ ln x \; \ mathrm {d} x = {\ frac {1} {c}} \ left (e ^ {cx} \ log | x | - \ int {\ frac {e ^ {cx} \ mathrm {d} x} {x}} \ right)}

\int e^{cx}\sin bx\;\mathrm {d} x={\frac {e^{cx}}{c^{2}+b^{2}}}(c\sin bx-b\cos bx)

{\ displaystyle \ int e ^ {cx} \ sin bx \; \ mathrm {d} x = {\ frac {e ^ {cx}} {c ^ {2} + b ^ {2}}} (c \ sin bx-b \ cos bx)}

{\ displaystyle \ int e ^ {cx} \ sin bx \; \ mathrm {d} x = {\ frac {e ^ {cx}} {c ^ {2} + b ^ {2}}} (c \ sin bx-b \ cos bx)}

\int e^{cx}\cos bx\;\mathrm {d} x={\frac {e^{cx}}{c^{2}+b^{2}}}(c\cos bx+b\sin bx)

{\ displaystyle \ int e ^ {cx} \ cos bx \; \ mathrm {d} x = {\ frac {e ^ {cx}} {c ^ {2} + b ^ {2}}} (c \ cos bx + b \ sin bx)}

{\ displaystyle \ int e ^ {cx} \ cos bx \; \ mathrm {d} x = {\ frac {e ^ {cx}} {c ^ {2} + b ^ {2}}} (c \ cos bx + b \ sin bx)}

\int e^{cx}\sin ^{n}x\;\mathrm {d} x={\frac {e^{cx}\sin ^{n-1}x}{c^{2}+n^{2}}}(c\sin x-n\cos x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\sin ^{n-2}x\;\mathrm {d} x

{\ displaystyle \ int e ^ {cx} \ sin ^ {n} x \; \ mathrm {d} x = {\ frac {e ^ {cx} \ sin ^ {n-1} x} {c ^ {2 } + n ^ {2}}} (c \ sin xn \ cos x) + {\ frac {n (n-1)} {c ^ {2} + n ^ {2}}} \ int e ^ {cx } \ sin ^ {n-2} x \; \ mathrm {d} x}

{\ displaystyle \ int e ^ {cx} \ sin ^ {n} x \; \ mathrm {d} x = {\ frac {e ^ {cx} \ sin ^ {n-1} x} {c ^ {2 } + n ^ {2}}} (c \ sin xn \ cos x) + {\ frac {n (n-1)} {c ^ {2} + n ^ {2}}} \ int e ^ {cx } \ sin ^ {n-2} x \; \ mathrm {d} x}

\int e^{cx}\cos ^{n}x\;\mathrm {d} x={\frac {e^{cx}\cos ^{n-1}x}{c^{2}+n^{2}}}(c\cos x+n\sin x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\cos ^{n-2}x\;\mathrm {d} x

{\ displaystyle \ int e ^ {cx} \ cos ^ {n} x \; \ mathrm {d} x = {\ frac {e ^ {cx} \ cos ^ {n-1} x} {c ^ {2 } + n ^ {2}}} (c \ cos x + n \ sin x) + {\ frac {n (n-1)} {c ^ {2} + n ^ {2}}} \ int e ^ {cx} \ cos ^ {n-2} x \; \ mathrm {d} x}

{\ displaystyle \ int e ^ {cx} \ cos ^ {n} x \; \ mathrm {d} x = {\ frac {e ^ {cx} \ cos ^ {n-1} x} {c ^ {2 } + n ^ {2}}} (c \ cos x + n \ sin x) + {\ frac {n (n-1)} {c ^ {2} + n ^ {2}}} \ int e ^ {cx} \ cos ^ {n-2} x \; \ mathrm {d} x}

\int {1 \over \sigma {\sqrt {2\pi }}}\,e^{-{(x-\mu )^{2}/2\sigma ^{2}}}\;={\mbox{erf}}\,{\frac {x-\mu }{\sigma {\sqrt {2}}}}

{\ displaystyle \ int {1 \ over \ sigma {\ sqrt {2 \ pi}}} \, e ^ {- {(x- \ mu) ^ {2} / 2 \ sigma ^ {2}}} \; = {\ mbox {erf}} \, {\ frac {x- \ mu} {\ sigma {\ sqrt {2}}}}}

{\ displaystyle \ int {1 \ over \ sigma {\ sqrt {2 \ pi}}} \, e ^ {- {(x- \ mu) ^ {2} / 2 \ sigma ^ {2}}} \; = {\ mbox {erf}} \, {\ frac {x- \ mu} {\ sigma {\ sqrt {2}}}}}

Bibliografie

Murray R. Spiegel, Manual de matematică , Etas Libri, 1974, p. 85.

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Tabelele integrale