Solved: "Evaluate the indefinate integral. ∫x2sin⁡(x3)dx"

skomminbv

Answered question

2021-11-22

Evaluate the indefinate integral.

\int x^{2} \sin (x^{3}) d x

Glenn Cooper

Beginner2021-11-23Added 12 answers

Step 1
Evaluate indefinite integral.

\int x^{2} \sin (x^{3}) d x

Consider,

u = x^{3}

d u = 3 x^{2} d x

\frac{d u}{3} = x^{2} d x

Step 2
Substitute the values in integral and evaluate.

\int \sin (u) \frac{d u}{3}

\Rightarrow \frac{1}{3} \int \sin u d u

\Rightarrow \frac{1}{3} (- \cos u)

Substitute u value,

\Rightarrow \frac{1}{3} (- {\cos (x)}^{3}) + C

\Rightarrow - \frac{1}{3} {\cos (x)}^{3} + C

kayleeveez7

Beginner2021-11-24Added 10 answers

Step 1: Use Integration by Substitution.
Let

u = x^{3}, d u = 3 x^{2} d x, t h e n x^{2} d x = \frac{1}{3} d u

Step 2: Using u and du above, rewrite

\int x^{2} \sin (x^{3}) d x

\int \frac{\sin u}{3} d u

Step 3: Use Constant Factor Rule:

\int c f (x) d x = c \int f (x) d x

\frac{1}{3} \int \sin u d u

Step 4: Use Trigonometric Integration: the integral of

\sin u i s - \cos u

- \frac{\cos u}{3}

Step 5: Substitute

u = x^{3}

back into the original integral.

- \frac{\cos (x^{3})}{3}

Step 6: Add constant.

- \frac{\cos (x^{3})}{3} + C

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