& _1^3 x^2 x d x= _1^3( x) x^2 d x = & [( x) x^2 d x ]_1^3- _1^3 [d d x ( x) x^2 d x ] d x = & [( x) (x^3 3 ) ]_1^3- _1^3 1 x x^3 3 d x = & 1 3 [x^3 x ]_1^3-1 3 _1^3 x^2 d x = & 1 3 [27 3-0]-1 3 [x^3 3 ]_1^3 [ 1=0] = & 9 3-1 9 (27-1) = & 9 3-26 9 .

_1^3 x^2 x d x

Download App

Question

$\int_1^3 x^2 \log x d x$

✓

Answer

$
\begin{aligned}
& \int_1^3 x^2 \log x d x=\int_1^3(\log x) \cdot x^2 d x \\
= & {\left[(\log x) \int x^2 d x\right]_1^3-\int_1^3\left[\frac{d}{d x}(\log x) \int x^2 d x\right] d x } \\
= & {\left[(\log x)\left(\frac{x^3}{3}\right)\right]_1^3-\int_1^3 \frac{1}{x} \times \frac{x^3}{3} d x } \\
= & \frac{1}{3}\left[x^3 \log x\right]_1^3-\frac{1}{3} \int_1^3 x^2 d x \\
= & \frac{1}{3}[27 \log 3-0]-\frac{1}{3}\left[\frac{x^3}{3}\right]_1^3 \quad \ldots[\because \log 1=0] \\
= & 9 \log 3-\frac{1}{9}(27-1) \\
= & 9 \log 3-\frac{26}{9} .
\end{aligned}
$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(2 Marks)" from Definite Integration (p-1)→Definite Integration (p-1) — all question types→Maths (commerce) STD 12 Commerce / Arts question papers→Maharashtra Board STD 12 Commerce / Arts question papers→Maharashtra Board question paper generator→

Definite Integration (p-1) — Maths (commerce) STD 12 Commerce / Arts — Question

Answer

Need a full question paper?

Explore more

Similar questions