Question

Integration problem

Answer 1

$\bold{ \underline{ \underline{ \pink{answer}}}} \longrightarrow \\ 1) \green{ \dfrac{5}{3} {x}^{3} - 4 {x}^{2} + 5x + c} \\ 2) \blue{8 logx + \dfrac{5}{x} + - \dfrac{3}{ {x}^{2} } + c} \\ 3) \green{ \dfrac{2}{3} {x}^{ \frac{3}{2} } + \dfrac{2}{3} {x}^{ \frac{1}{2} } + c} \\ 4) \blue{\dfrac{4}{5} {t}^{5} + t - \dfrac{4}{3} {t}^{3} + c} \\ 5) \green{5sin \theta \: + c} \\ 6) \blue{log|sec x| + 7 {e}^{x} + c} \\ \bold{ \underline{ \underline{ \pink{concept \: used}}}} \longrightarrow \\ 1) \: \orange{\displaystyle \int {x}^{n} dx = \dfrac{ {x}^{n + 1} }{n + 1} + c }\\ 2) \orange{\displaystyle \int \dfrac{1}{x} dx = log |x| + c} \\ 3) \orange{\displaystyle \int \: cos \theta \: d \theta = sin \theta \: + c} \\ 4) \orange{ \displaystyle \int \: tanx \: dx \: = log |secx| + c} \\ 5) \orange{\displaystyle \int \: {e}^{x} dx = {e}^{x} + c}$

$\bold{ \underline{ \underline{ \pink{solution}}}}\longrightarrow \\ 1) \displaystyle \int \: (5 {x}^{2} - 8x + 5) \: dx \\ = 5 \displaystyle \int {x}^{2} dx - 8 \displaystyle \int \: x \: dx \: + 5 \displaystyle \int \: 1 \: dx \\ = 5 \dfrac{ {x}^{3 + 1} }{3 + 1} - 8 \dfrac{ {x}^{1 + 1} }{1 + 1} + 5x + c \\ = \dfrac{5}{3} {x}^{3} - 4 {x}^{2} + 5x + c$

$2) \displaystyle \int \: ( \dfrac{8}{x} - \dfrac{5}{ {x}^{2} } + \dfrac{6}{ {x}^{3} } )dx \\ = 8 \displaystyle \int \: \dfrac{1}{x} dx - 5 \displaystyle \int \: {x}^{ - 2} dx + 6 \displaystyle \int \: {x}^{ - 3} dx \\ = 8 \: log |x| - 5 \dfrac{ {x}^{ - 2 + 1} }{ - 2 + 1} + 6 \dfrac{ {x}^{ - 3 + 1} }{ - 3 + 1} + c \\ = 8log |x| - 5 \dfrac{ {x}^{ - 1} }{ - 1} + 6 \dfrac{ {x}^{ - 2} }{ - 2} + c \\ = 8log |x| + \dfrac{5}{x} - \dfrac{3}{ {x}^{2} } + c$

$3) \displaystyle \int( \sqrt{x} + \dfrac{3}{ \sqrt{x} } )dx \\ = \displaystyle \int \: {x}^{ \frac{1}{2} } dx \: + 3 \displaystyle \int \: {x}^{ - \frac{1}{2} } dx \\ = \dfrac{ {x}^{ \frac{1}{2} + 1 } }{ \frac{1}{2} + 1} + \dfrac{1}{3} \: \dfrac{ {x}^{ - \frac{1}{2} + 1} }{ (- \frac{1}{2} + 1)} + c \\ = \dfrac{2}{3} {x}^{ \frac{3}{2} } + \dfrac{1}{3} \dfrac{ {x}^{ \frac{1}{2} } }{ \dfrac{1}{2} } + c \\ = \dfrac{2}{3} {x}^{ \frac{3}{2} } + \dfrac{2}{3} {x}^{ \frac{1}{2} } + c$

$4) \displaystyle \int \: {(2 {t}^{2} - 1)}^{2} dt \\ = \displaystyle \int(4 {t}^{4} + 1 - 4 {t}^{2} )dt \\ = 4 \displaystyle \int {t}^{4} dt + \displaystyle \int1dt \: - 4 \displaystyle \int {t}^{2} dt \\ = 4 \dfrac{ {t}^{4 + 1} }{4 + 1} + t - \dfrac{4}{3} {t}^{3} + c \\ = \dfrac{4}{5} {t}^{5} + t - \dfrac{4}{3} {t}^{3} + c$

$5) \displaystyle \int5cos \theta \: d \theta \\ = 5 \: sin \theta \: + c$

$6) \displaystyle \int \: (tanx + 7 {e}^{x} )dx \\ = \displaystyle \int \: tanx \: dx + 7 \displaystyle \int \: {e}^{x} dx \\ = log |secx| + 7 \: {e}^{x} + c$

Answer 2

Answer:

i don't know coz i m not too intelligent like the first person who answered this question i.e. rishu bhai