Question

Ajudo em Calculo I!!

Answer 1

Como são muitas integrais, farei apenas o cálculo sem a explicação sobre cada uma delas.

$1) \int 2x + \cos(x) \: dx \\ \int 2x \: dx + \int \cos(x)dx \\ x {}^{2} + \sin(x) + k,k\in \mathbb{R}$

$2) \int x {}^{3} - 4 \cos(x) \: dx \\ \int x {}^{3} dx - 4 \int \cos(x)dx \\ \frac{x {}^{4} }{4} - 4 \sin(x) + k,k\in \mathbb{R}$

$3) \int \frac{1}{x {}^{3} } dx \\ \int {x}^{ - 3} dx = - \frac{1 }{ 2x {}^{2} } + k ,k\in \mathbb{R}$

$4) \int( 3x {}^{2} + 1 + 2x)dx \\ 3\int x {}^{2} dx+ \int 1dx +2 \int x \: dx \\ x {}^{3} + x + x {}^{2} + k,k\in \mathbb{R}$

$5) \int \sin(x) + \frac{ \tg(x)}{ \cos(x)} dx \\ \int \sin(x)dx + \int \frac{ \tg(x)}{ \cos(x)}dx \\ - \cos(x) + \int \frac{ \frac{ \sin(x)}{ { \cos(x)} }}{ { \cos(x)}} dx \\ - \cos(x) + \int \frac{ \sin(x)}{ \cos(x)} . \frac{1}{ \cos(x)} dx \\ - \cos(x) + \int \tg(x). \sec(x)dx \\ - \cos(x) + \sec(x) + k,k\in \mathbb{R}$

$6) \int 7 \sqrt{x {}^{5} } dx \\ 7 \int (x {}^{5} ) {}^{ \frac{1}{2} } dx \\ 7 \int x {}^{ \frac{5}{2} } dx \: \: \: \: \\ 7 . \frac{x {}^{ \frac{5}{2} + 1} }{ \frac{5}{2} + 1} + k \\ \frac{14x {}^{ \frac{7}{2} } }{7} \\ 2x^{ \frac{7}{2} } + k,k\in \mathbb{R}$

$7) \int x {}^{3} . \sqrt{x} dx \\ \int x {}^{3} .x {}^{ \frac{1}{2} } dx \\ \int x {}^{3 + \frac{1}{2} } dx \\ \int x {}^{ \frac{7}{2} } dx \\ \frac{2x {}^{ \frac{9}{2} } }{9} + k,k\in \mathbb{R}$

$8)\int \frac{dx}{ \sin {}^{2}(x) } \\ \int \frac{1}{ \sin {}^{2} (x)} dx \\ \int \csc {}^{2} (x)dx \\ - \cotg(x) + k,k\in \mathbb{R}$

Espero ter ajudado

Answer 2

Olá, siga a explicação:

1)

$\mathrm{Aplicar\:a\:regra\:da\:soma:} \\ \\ \boxed {\mathrm { \displaystyle \int\limits f(x) \pm g(x) ~dx= \displaystyle \int\limits f(x) ~ dx \pm \displaystyle \int\limits g(x) ~ dx }} \\\\\mathrm { \displaystyle \int\limits 2xdx + \displaystyle \int\limits cos(x) ~ dx } \\ \\\boxed { \mathrm { \displaystyle \int\limits 2xdx = x^2}} \\ \\\boxed {\mathrm {\displaystyle \int\limits cos(x) ~ dx= sen(x) }} \\ \\ \\\boxed { \mathrm { = \bf x^2 + sen(x) + C }}$

2)

$\mathrm{Aplicar\:a\:regra\:da\:soma:} \\ \\\mathrm { \displaystyle \int\limits f(x) \pm g(x) ~dx= \displaystyle \int\limits f(x) ~ dx \pm \displaystyle \int\limits g(x) ~ dx } \\\\\\\mathrm { \displaystyle \int\limits x^3 dx- \displaystyle \int\limits 4cos(x) dx } \\ \\ \\\boxed { \mathrm {\displaystyle \int\limits x^3dx = \dfrac{x^4}{4} }} \\ \\ \\\boxed { \mathrm {\displaystyle \int\limits 4cos(x) dx = 4sen (x) }} \\ \\ \\\boxed { \mathrm { \bf \dfrac{x^4}{4} - 4sen(x) + C}}$

3)

$\mathrm { Aplicar ~ as ~propriedades~dos ~expoentes }: \\ \\ \boxed {\mathrm { \dfrac{1}{a^b} = a^{-b} } }$

$\mathrm {\dfrac{1}{x^3} = x^{-3} } \\ \\\mathrm { \displaystyle \int\limits x^{-3} dx} \\ \\ \\\mathrm {Aplicar ~ a ~regra ~ da ~ potencia}: \\ \\ \\ \boxed {\mathrm { \displaystyle \int\limits x^adx= \dfrac{x^{a+1}}{a+1} , \quad a \neq -1 }} \\ \\\mathrm { \dfrac{x^{-3+1}}{-3+1} } \\ \\ \\\mathrm {Simplifica}:\\ \\\mathrm {\dfrac{x^{-3+1}}{-3+1} : \quad -\dfrac{1}{2x^2} } \\ \\ \\\boxed { \mathrm { \bf\dfrac{-1}{2x^2}+C }}$

4)

$\mathrm {Aplicar\:a\:regra\:da\:soma:} \\ \\\mathrm { \displaystyle \int\limits f(x) \pm g(x) ~dx= \displaystyle \int\limits f(x) ~ dx \pm \displaystyle \int\limits g(x) ~ dx } \\ \\\mathrm {\displaystyle \int\limits 3x^2dx+ \displaystyle \int\limits 1dx + \displaystyle \int\limits 2xdx} \\ \\\mathrm { \displaystyle \int\limits 3x^2dx= x^3 } \\ \\\mathrm { \displaystyle \int\limits 1dx = x} \\ \\\mathrm { \displaystyle \int\limits 2xdx = x^2} \\ \\\boxed { \mathrm {\bf x^3+x + x^2 + C }}$

5)

$\mathrm{Aplicar\:a\:regra\:da\:soma}: \\ \\\boxed { \mathrm { \displaystyle \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx }}$

$\mathrm {\displaystyle \int\limits sen(x) = -cos(x)} \\ \\\mathrm { \displaystyle \int\limits \dfrac{tan(x)}{cos(x)} dx = sec(x) } \\ \\ \\\boxed {\mathrm { - cos(x) + sec(x) = sen(x)tan(x) }} \\ \\ \\\boxed {\mathrm { \bf sen(x)tan(x)+C}}$

6)

$\boxed { \mathrm { \displaystyle \int\limits a \cdot f(x)dx = a \cdot \displaystyle \int\limits f(x) dx}} \\ \\ \\\mathrm {7\cdot \displaystyle \int \sqrt{x^5}dx} \\ \\ \\\boxed {\mathrm {\sqrt{x^5}=x^{\frac{5}{2}},\:\quad \mathrm{\:assumindo\:que}\:x\ge 0}} \\ \\\mathrm {7\cdot \displaystyle \int \:x^{\frac{5}{2}}dx} \\ \\\mathrm{Aplicar\:a\:regra\:da\:potencia}: \mathrm {\quad \displaystyle \int x^adx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1 }$

$\mathrm {7\cdot \dfrac{x^{\frac{5}{2}+1}}{\frac{5}{2}+1 } : \quad 2x^{\frac {7}{2}}} \\ \\ \\\boxed {\mathrm { \bf2x^{\frac{7}{2}}+C}}$

7)

$\mathrm {Simplifica}: \\ \\\mathrm {x^3 \sqrt{x} = x^{\frac{7}{2}} } \\ \\\mathrm {Aplicar ~ a ~ regra ~ da ~ potencia}: \\\\\boxed { \mathrm { \displaystyle \int\limits x^adx = \dfrac{x^{a+1}}{a+1}, \quad a \neq -1 } } \\ \\ \\\mathrm { = \dfrac{x^{\frac {7}{2} +1 }}{\frac{7}{2} +1} }$

$\mathrm {Simplifica}: \\ \\\mathrm {\dfrac{x^{ \frac {7}{2} +1}}{\frac {7}{2} +1} : \quad \dfrac{2}{9} x^{\frac {9}{2}} } \\ \\ \\\boxed { \mathrm { \bf \dfrac{2}{9} x^{ \frac {9}{2} } + C }}$

8)

$\mathrm { Aplicar ~ as ~ regras ~ de ~ integracao}: \\ \\\boxed {\mathrm { \displaystyle \int\limits \dfrac{1}{sen^2 (x)} = dx = -cot (x)}} \\ \\ \\\boxed { \mathrm {\bf -cot (x) + C} }$

• Att. MatiasHP

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