Calculo Integral Ug 2021: INTEGRALES QUE INCLUYEN POTENCIAS DE SENO Y COSENO

Caso 1: Potencias pares de seno y coseno

Se aplica el seno y coseno del ángulo mitad

${sen^{2}\, x= \displaystyle\frac{1-cos\, 2x}{2}, \ \ \ \ \ \ cos^{2}\, x= \displaystyle\frac{1+cos\, 2x}{2}}$

Ejemplo:

${\displaystyle\int cos^{2}\, x \, dx}$

${\begin{array}{rcl} \displaystyle\int cos^{2}\, x \, dx & = & \displaystyle\int \left( \sqrt{\frac{1+cos\, 2x}{2}}\right)^{2} dx \\ &&\\ &=& \displaystyle\int \frac{1+cos\, 2x}{2}\, dx \\ && \\ &=& \displaystyle\frac{1}{2}\int (1+cos\, 2x)\, dx \\ && \\ &=& \displaystyle\frac{1}{2}\left(x + \frac{1}{2}sen\, 2x \right)+C \\ && \\ &=& \displaystyle\frac{1}{2}x + \frac{1}{4}sen\, 2x+C \end{array}}$

Ejemplo:

${\displaystyle\int sen^{4}\, x \, dx}$

${\begin{array}{rcl} \displaystyle\int sen^{4}\, x \, dx & = & \displaystyle\int \left( \frac{1-cos\, 2x}{2}\right)^{2} dx \\ &&\\ &=& \displaystyle\int \frac{1-2cos\, 2x + cos^{2} 2x}{4}\, dx \\ && \\ &=& \displaystyle\frac{1}{4}\int dx - \frac{2}{4}\int cos\, 2x \, dx +\frac{1}{4}\int cos^{2} 2x \, dx \\ && \\ &=& \displaystyle\frac{1}{4}x-\frac{1}{4}sen\, 2x + \frac{1}{4} \int \frac{1+cos\, 4x}{2}\,dx \\ && \\ &=&\displaystyle\frac{1}{4}x-\frac{1}{4}sen\, 2x + \frac{1}{8}x + \frac{1}{32}sen\, 4x +C \\ && \\ &=&\displaystyle\frac{3}{8}x-\frac{1}{4}sen\, 2x + \frac{1}{8}x + \frac{1}{32}sen\, 4x +C \end{array}}$

Caso 2: Potencias impares de seno y coseno

Se relacionan con la identidad trigonométrica

${sen^{2} x + cos^{2} x = 1}$

Ejemplo:

${\displaystyle\int sen^{3}\, x \, dx}$

${\begin{array}{rcl} \displaystyle\int sen^{3}\, x \, dx & = & \displaystyle\int sen\, x \, sen^{2}x\, dx \\ &&\\ &=& \displaystyle\int sen\, x (1-cos^{2}x)\, dx \\ && \\ &=& \displaystyle\int \left( sen\, x-cos^{2}x\, sen\, x\right) \, dx \\ && \\ &=& -cos\, x+\displaystyle\frac{1}{3}cos^{3}x+C \end{array}}$

Ejemplo:

${\displaystyle\int cos^{3}\, x \, dx}$

${\begin{array}{rcl} \displaystyle\int cos^{3}\, x \, dx & = & \displaystyle\int cos\, x \, cos^{2}x\, dx \\ &&\\ &=& \displaystyle\int cos\, x (1-sen^{2}x)\, dx \\ && \\ &=& \displaystyle\int \left(cos\, x-sen^{2}x\, cos\, x\right) \, dx \\ && \\ &=& sen\, x-\displaystyle\frac{1}{3}sen^{3}x+C \end{array}}$

Ejemplo:

${\displaystyle\int cos^{3}\, x \, dx}$

${\begin{array}{rcl} \displaystyle\int cos^{5}\, x \, dx & = & \displaystyle\int cos\, x \, cos^{4}x\, dx \\ &&\\ &=& \displaystyle\int cos\, x (1-sen^{2}x)^{2}\, dx \\ && \\ &=& \displaystyle\int cos\, x \, dx -2\int sen^{2}x\, cos\, x \, dx + \int sen^{4}x\, cos\, x\, dx \\ && \\ &=& sen\, x-\displaystyle\frac{2}{3}sen^{3}x+\frac{1}{5}sen^{5}x+C \end{array}}$

Caso 3: Con exponente par e impar

El exponente impar se transforma en uno par y otro impar.

Ejemplo:

${\displaystyle\int sen^{5}x\, cos^{2}\, x \, dx}$

${\begin{array}{rcl} \displaystyle\int sen^{5}x\, cos^{2} x \, dx & = & \displaystyle\int sen\, x \, sen^{4}x\, cos^{2}x\, dx \\ &&\\ &=& \displaystyle\int \left(1-cos^{2}x\right)^{2}\, sen\, x\, cos^{2} x \,dx \\ && \\ &=& \displaystyle\int \left(1-2cos^{2} x+cos^{4}x\right)\, sen\, x \, cos^{2} x\, dx \\ && \\ &=& \displaystyle\int \left(cos^{2} x\, sen\, x-2cos^{4}x\, sen\, x+cos^{6}x\, sen\, x\right)\, dx \\ && \\ &=& -\displaystyle\frac{1}{3}cos^{3}x+\frac{2}{5}cos^{5}x-\frac{1}{7}cos^{7}x+C \end{array}}$

Ejemplo:

${\displaystyle\int sen^{4}x\, cos\, x \, dx}$

${\displaystyle\int sen^{4}x\, cos\, x \, dx & = & \displaystyle\frac{1}{5} sen^{5} x+C}$

Ejemplo:

${\displaystyle\int cos^{2}x\, sen^{3} x \, dx}$

${\begin{array}{rcl} \displaystyle\int cos^{2}x\, sen^{3} x \, dx & = & \displaystyle\int cos^{2} x \, sen\, x\, sen^{2}x\, dx \\ &&\\ &=& \displaystyle\int cos^{2} x \, sen\, x\left(1-cos^{2}x\right) \,dx \\ && \\ &=& \displaystyle\int \left(cos^{2} x \, sen\, x-cos^{4} x \, sen\, x\right)\, dx \\ && \\ &=& -\displaystyle\frac{1}{3}cos^{3}x+\frac{1}{5}cos^{5}x+C \end{array}}$

Ejemplo:

${\displaystyle\int \frac{sen^{3} x}{cos\, x} \, dx}$

${\begin{array}{rcl} \displaystyle\int \frac{sen^{3} x}{cos\, x} \, dx & = & \displaystyle\int \frac{sen\, x\, sen^{2} x}{cos\, x} \, dx \\ &&\\ &=& \displaystyle\int \frac{sen\, x\, (1-cos^{2} x)}{cos\, x} \, dx \\ && \\ &=& \displaystyle\int \frac{sen\, x}{cos\, x}\, dx - \int sen\,x \, cos\, x\, dx \\ && \\ &=& -ln(cos\, x)+\displaystyle\frac{1}{2}cos^{2}x+C \end{array}}$

Caso 4: Productos de tipo sen(nx) · cos(mx)

Se transforman los productos en sumas:

${sen\, A\cdot cos\, B=\displaystyle\int \frac{1}{2}[sen(A+B)+sen(A-B)]}$

${cos\, A\cdot sen\, B=\displaystyle\int \frac{1}{2}[sen(A+B)-sen(A-B)]}$

${cos\, A\cdot cos\, B=\displaystyle\int \frac{1}{2}[cos(A+B)+cos(A-B)]}$

${sen\, A\cdot sen\, B=\displaystyle\int \frac{1}{2}[cos(A+B)-cos(A-B)]}$

Ejemplo:

${\displaystyle\int sen\, 3x\, cos\, 2x \, dx}$

${\begin{array}{rcl} \displaystyle\int sen\, 3x\, cos\, 2x \, dx & = & \displaystyle\frac{1}{2}\int (sen\, 5x + sen\, x)\, dx \\ &&\\ &=& \displaystyle\frac{1}{2}\left(-\frac{cos\, 5x}{5}-cos\, x\right)+C \end{array}}$

Ejemplo:

${\displaystyle\int cos\, 5x\, sen\, 3x \, dx}$

${\begin{array}{rcl} \displaystyle\int cos\, 5x\, sen\, 3x \, dx & = & \displaystyle\frac{1}{2}\int (sen\, 8x - sen\, 2x)\, dx \\ &&\\ &=& -\displaystyle\frac{1}{16}cos\, 8x+\frac{1}{4}cos\, 2x+C \end{array}}$

Ejemplo:

${\displaystyle\int cos\, 4x\, cos\, 2x \, dx}$

${\begin{array}{rcl} \displaystyle\int cos\, 4x\, cos\, 2x \, dx & = & \displaystyle\frac{1}{2}\int (cos\, 6x + cos\, 2x)\, dx \\ &&\\ &=& \displaystyle\frac{1}{12}sen\, 6x+\frac{1}{4}sen\, 2x+C \end{array}}$

Ejemplo:

${\displaystyle\int sen\, 3x\, sen\, 7x \, dx}$

${\begin{array}{rcl} \displaystyle\int sen\, 3x\, sen\, 7x \, dx & = & -\displaystyle\frac{1}{2}\int (cos\, 10x - cos\, (-4x)\, dx \\ &&\\ &=& -\displaystyle\frac{1}{2}\int cos\, 10x\, dx+\frac{1}{2}\int cos\, 4x\, dx \\ &&\\ &=& -\displaystyle\frac{1}{20}sen\, 10x+\frac{1}{8}sen\, 4x +C \end{array}}$