Find the indefinite integral. ∫x2−2xx3−3x2dx

Question

Find the indefinite integral.   ∫x2−2xx3−3x2dx

Marian Tucker · Accepted Answer

Step 1  Given integral:  ∫x2−2xx3−3x2dx  Step 2  Substitute x3−3x2=t and differentiate it w.r.t x  ddx(x3−3x2)=dtdx  3x2−6x=dtdx  (x2−2x)dx=dt3  So,  ⇒13tdt  ⇒13∫1tdt  We know that ∫1xdx=log⁡(x)+C  Where, &quot;C&quot; is integrating constant.  ⇒13log⁡(t)+C  Step 3  Substitute x3−3x2=t, we get  ⇒13log⁡(x3−3x2)+C

Royce Moore · Accepted Answer

Step 1: Use Integration by Substitution.
Let

u = x^{3} - 3 x^{2}, d u = 3 x^{2} - 6 x d x, t h e n x^{2} - 2 x d x = \frac{1}{3} d u

Step 2: Using u and du above, rewrite

\int \frac{x^{2} - 2 x}{x^{3} - 3 x^{2}} d x

.

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

Step 3: Use Constant Factor Rule:

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

.

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

Step 4: The derivative of

\ln x i s \frac{1}{x}

.

\frac{\ln u}{3}

Step 5: Substitute

u = x^{3} - 3 x^{2}

back into the original integral.

\frac{\ln (x^{3} - 3 x^{2})}{3}

Step 6: Add constant.

\frac{\ln (x^{3} - 3 x^{2})}{3} + C

Find the indefinite integral. \int \frac{x^{2}-2x}{x^{3}-3x^{2}}dx

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