Question

Calculate the following integral ∫(3x^-7 + x^-1)dx

Answer 1

Answer:

∫(3x^-7 + x^-1)dx = 3∫x^-7 dx + ∫ x^-1 dx

=> - 3 x^-6 / 6 + x^0 /0 + C

=> - x^(-6)/ 2 + C

Answer 2

Question:

Evaluate:

$\int(3 {x}^{ - 7} + {x}^{ - 1}) dx$

Answer:

Take

$i = \int(3 {x}^{ - 7} + {x}^{ - 1} )dx \\ \\ i = 3 \int \: {x}^{ - 7} dx + \int \: \frac{1}{x} dx \\ \\ integrate \: w.r.t.x\\ \\ i = 3 \: \frac{ {x}^{ - 7 + 1} }{ - 7 + 1} + log |x| + c \\ \\ i = 3( \frac{ {x}^{ - 6} }{ - 6} ) + log |x| + c \\ \\ i = \frac{ - 3}{6} (\frac{1}{ {x}^{6} } ) + log |x| + c \\ \\ i = - \frac{1}{2 {x}^{6} } + log |x| + c$

Formulae used:

$\star \int {x}^{n} dx = \frac{ {x}^{n + 1} }{n + 1} + c\\ \\ \star \int \frac{1}{x} = log |x| + c$