Question

integrate sin ( x)³. please answer the question ​

Answer 1

Answer:

∫sin(x3)dx≈x44−x1060+x141920

Step-by-step explanation:

Start with the known formula for sinx and then replace x by x3

sinx=∑∞n=0(−1)nx2n+1(2n+1)!

sin(x3)=∑∞n=0(−1)n(x3)2n+1(2n+1)!

The integral becomes ∫∑∞n=0(−1)nx6n+3(2n+1)!dx=∑∞n=0(−1)nx6n+4(2n+1)!×(6n+4)+C

For small values of x, the first few terms can be taken to give reasonable accuracy in case you want to evaluate a definite integral. The more terms you take the more accurate the answer is.

∫sin(x3)dx≈x44−x1060+x141920

Hope this helps.

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Answer 2

${} \bf{ \huge{ \boxed{ \underline{ \mathfrak{ \blue{Answer:-}}}}}}$

$\color{Red}{Step 1:-}$$\displaystyle \int {sin}^{3} x \: dx$

$\color{Red}{Step 2:-}\\ Use \: Pythagorean \: Identities: \\ sin ^{2} x =( 1 - cos ^{2} )sin \: x \: dx \\ \displaystyle \int(1 - {cos}^{2} x)sin \: x \: dx.$

$\color{Red}{Step 3:-}\\Use \: Integration \: by \: Substitution .\\ Let \: u = cos \: x \: du = - sin \:x \: dx.$

$\color{Red}{Step 4:-}\\Using \: u \: and \: du, \: above, \: rewrite \\ \displaystyle \int(1 - {cos}^{2} x)sin \: x \: d x.\\ \displaystyle \int - (1 - {u}^{2} )du$

$\color{Red}{Step 5:-}$$Use \: Power \: Rule \: \displaystyle \int \: {x}^{n \:} dx = \frac{ {x}^{n + 1} }{n + 1} + C \\ \frac{ {u}^{3} }{3 } - u$

$\color{Red}{Step 6:-}$$Substitute \: u = cos \: x \: back \: into \: the \: original \: integral .\\ \frac{cos ^{3} x}{3} - cos \: x$

$\color{Red}{Step 7:-}$$\\Add \: constant \\ \frac{cos \: x ^{3} }{3} - cos \: x + C$