Integral by substitution

Calculus Level 1

Determine the indefinite integral of the following expression: $\cos { \frac{x}{5} } \sin^6 { \frac{x}{5}}.$

Details and assumptions

Use $C$ as the constant of integration.

\frac{3}{7} \sin ^7 { \frac {x}{2}} + C

\frac{5}{7} \sin ^7 { \frac {x}{5}} + C

\frac{2}{7} \sin ^7 { \frac {x}{3}} + C

\frac{6}{7} \sin ^7 { \frac {x}{4}} + C

1 solution

Oon Han
Dec 5, 2017

Let $u=\sin { \frac { x }{ 5 } }$ . Then $du=\frac { 1 }{ 5 } \cos { \frac { x }{ 5 } } dx$ and $dx=\frac { 1 }{ \frac { 1 }{ 5 } \cos { \frac { x }{ 5 } } } du$ .

$\int { \cos { \frac { x }{ 5 } } { \sin ^{ 6 }{ \frac { x }{ 5 } } } } dx=\int { \frac { \cos { \frac { x }{ 5 } } }{ \frac { 1 }{ 5 } \cos { \frac { x }{ 5 } } } { u }^{ 6 } } du=5\times \int { { u }^{ 6 } } du=5\frac { { u }^{ 7 } }{ 7 } +C=\frac { 5 }{ 7 } \sin ^{ 7 }{ \frac { x }{ 5 } } +C$

Integral by substitution

1 solution

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