Answer:
Answer is on the pic...
Sorry just little bad handwriting.
Thanks...
Question:- Integrate the following
[tex]\qquad\qquad\sf \: \dfrac{1}{ \sqrt[4]{ {x}^{7} } } \\ \\ [/tex]
[tex] \boxed{ \sf{ \:\sf \: \displaystyle\int\sf \dfrac{1}{ \sqrt[4]{ {x}^{7} } } \: dx \: = \: - \dfrac{4}{3} \bigg(x\bigg)^{\frac{ -3}{4}} + c \: = \: - \: \dfrac{4}{3 \sqrt[4]{ {x}^{3} } } + c \:}} \\ \\ [/tex]
Step-by-step explanation:
Given integral is
[tex]\sf \: \displaystyle\int\sf \dfrac{1}{ \sqrt[4]{ {x}^{7} } } \: dx \\ \\ [/tex]
can be rewritten as
[tex]\sf \: = \: \displaystyle\int\sf \dfrac{1}{\bigg(x\bigg)^{ \frac{7}{4} } } \: dx \\ \\ [/tex]
[tex]\sf \: = \: \displaystyle\int\sf {\bigg(x\bigg)^{ - \frac{7}{4} } } \: dx \\ \\ [/tex]
So, using this result we get
[tex]\sf \: = \: \dfrac{\bigg(x\bigg)^{ - \frac{7}{4} + 1} }{ - \dfrac{7}{4} + 1} + c \: \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{\bigg(x\bigg)^{\frac{ - 7 + 4}{4}} }{\dfrac{ - 7 + 4}{4}} + c \: \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{\bigg(x\bigg)^{\frac{ -3}{4}} }{\dfrac{ - 3}{4}} + c \: \\ \\ [/tex]
[tex]\sf \: = \: - \dfrac{4}{3} \bigg(x\bigg)^{\frac{ -3}{4}} + c \: \\ \\ [/tex]
Hence,
[tex]\sf\implies \sf \: \displaystyle\int\sf \dfrac{1}{ \sqrt[4]{ {x}^{7} } } \: dx = \: - \dfrac{4}{3} \bigg(x\bigg)^{\frac{ -3}{4}} + c \: \\ \\ [/tex]
Or
[tex]\sf\implies \sf \: \displaystyle\int\sf \dfrac{1}{ \sqrt[4]{ {x}^{7} } } \: dx = \: - \: \dfrac{4}{3 \sqrt[4]{ {x}^{3} } } + c \: \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information
[tex]\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}[/tex]
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Answers & Comments
Verified answer
Answer:
Answer is on the pic...
Sorry just little bad handwriting.
Thanks...
Question:- Integrate the following
[tex]\qquad\qquad\sf \: \dfrac{1}{ \sqrt[4]{ {x}^{7} } } \\ \\ [/tex]
Answer:
[tex] \boxed{ \sf{ \:\sf \: \displaystyle\int\sf \dfrac{1}{ \sqrt[4]{ {x}^{7} } } \: dx \: = \: - \dfrac{4}{3} \bigg(x\bigg)^{\frac{ -3}{4}} + c \: = \: - \: \dfrac{4}{3 \sqrt[4]{ {x}^{3} } } + c \:}} \\ \\ [/tex]
Step-by-step explanation:
Given integral is
[tex]\sf \: \displaystyle\int\sf \dfrac{1}{ \sqrt[4]{ {x}^{7} } } \: dx \\ \\ [/tex]
can be rewritten as
[tex]\sf \: = \: \displaystyle\int\sf \dfrac{1}{\bigg(x\bigg)^{ \frac{7}{4} } } \: dx \\ \\ [/tex]
[tex]\sf \: = \: \displaystyle\int\sf {\bigg(x\bigg)^{ - \frac{7}{4} } } \: dx \\ \\ [/tex]
So, using this result we get
[tex]\sf \: = \: \dfrac{\bigg(x\bigg)^{ - \frac{7}{4} + 1} }{ - \dfrac{7}{4} + 1} + c \: \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{\bigg(x\bigg)^{\frac{ - 7 + 4}{4}} }{\dfrac{ - 7 + 4}{4}} + c \: \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{\bigg(x\bigg)^{\frac{ -3}{4}} }{\dfrac{ - 3}{4}} + c \: \\ \\ [/tex]
[tex]\sf \: = \: - \dfrac{4}{3} \bigg(x\bigg)^{\frac{ -3}{4}} + c \: \\ \\ [/tex]
Hence,
[tex]\sf\implies \sf \: \displaystyle\int\sf \dfrac{1}{ \sqrt[4]{ {x}^{7} } } \: dx = \: - \dfrac{4}{3} \bigg(x\bigg)^{\frac{ -3}{4}} + c \: \\ \\ [/tex]
Or
[tex]\sf\implies \sf \: \displaystyle\int\sf \dfrac{1}{ \sqrt[4]{ {x}^{7} } } \: dx = \: - \: \dfrac{4}{3 \sqrt[4]{ {x}^{3} } } + c \: \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information
[tex]\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}[/tex]