Integration by Parts

Integration by Parts

General Formula

$$ \int u , dv = uv - \int v , du $$

Reduction Formula

Show that:

$$ \int x^m e^{ax} , dx = \frac{x^m e^{ax}}{a} - \frac{m}{a} \int x^{m-1} e^{ax} , dx $$

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Tutorial Problems

1. Using the Reduction Formula

Evaluate the following integrals:

• 1(a) $\int x^3 e^{2x} , dx$

• 1(b) $\int x^2 (e^x - 1) , dx$

• 1(c) $\int x e^x , dx$

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2. Integration by Parts

Integrate the following:

• 2(a) $\int \ln x , dx$

• 2(b) $\int \ln x^2 , dx$

• 2(c) $\int (\ln x)^2 , dx$

• 2(d) $\int x^2 \sin x , dx$

• 2(e) $\int e^x \sin 2x , dx$

• 2(f) $\int x^3 e^x , dx$

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3. Identifying Methods

Identify what method(s) are possible to find:

$$ \int x \sqrt{3x + 7} , dx $$

Then evaluate the integral.

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4. Integration by Parts

Solve the following using integration by parts:

• 4(a) $\int x \sqrt{1 + x} , dx$

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

• 4(c) $\int \ln(x^2 + 1) , dx$

• 4(d) $\int e^x \cos x , dx$

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Source: Soalan Tut 2 FAD1014(25.26)..pdf