Phrases Previous Year Questions (PYQs) – Indefinite Integration – Page 1 of 2

Quick Solution

Given:

$\int f(x)dx=g(x)$

Required: $\int x^{5}f(x^3)dx$

Use substitution:

Let $u=x^3\Rightarrow du=3x^{2}dx\Rightarrow dx=\frac{du}{3x^2}$

Now rewrite the integral:

$$\int x^{5}f(x^3)dx=\int x^{5}f(u)\cdot \frac{du}{3x^2}=\frac{1}{3}\int x^{3}f(u)du$$

But $x^3=u$, so:

$$\frac{1}{3}\int uf(u)du$$

Now integrate by parts or use the identity:

$$\int uf(u)du=ug(u)-\int g(u)du$$

Final answer:

$$\int x^{5}f(x^3)dx=\frac{1}{3}[x^{3}g(x^3)-\int g(x^3)\cdot 3x^{2}dx]=x^{3}g(x^3)-\int x^{2}g(x^3)dx$$

$$\boxed{{\displaystyle \int x^{5}f(x^3)dx=x^{3}g(x^3)-\int x^{2}g(x^3)dx}}$$