FAD1014: MATHEMATICS II — Tutorial 2

Centre for Foundation Studies in Science, University of Malaya
Session 2025/2026

Question 1: Integration by Parts Formula

(a) $\int x^3 e^{2x} dx$

(b) Show that $\int x^m e^{ax} dx = \frac{1}{a}x^m e^{ax} - \frac{m}{a}\int x^{m-1}e^{ax} dx$

Hence, evaluate the following:

(c) $\int x^2(e^x - 1)dx$

(d) $\int x e^x dx$

Integrate the following:

(a) $\int \ln x , dx$

(b) $\int \ln x^2 , dx$

(c) $\int(\ln x)^2 dx$

(d) $\int x^2 \sin x , dx$

(e) $\int e^x \sin 2x , dx$

(f) $\int x^3 e^x dx$

Identify what method(s) is(are) possible to be used to find $\int x\sqrt{3x + 7} , dx$, and then find it.

Solve the question below using integration by parts method:

(a) $\int x\sqrt{1 + x} , dx$

(b) $\int \frac{(\ln x)^2}{x} dx$

(c) $\int \ln(x^2 + 1) , dx$

(d) $\int e^x \cos x , dx$

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