3 Board Questions on Integration

This lesson works through three Class XII board questions on integration, each showing a standard technique you can reuse.

Question 1: an exponential integral

Evaluate

$\int \frac{x^2 + 1}{(x+1)^2}\, e^{x}\, dx.$

Whenever you see $e^{x}$ multiplying a fraction, look for the form

$\int \big(f(x) + f'(x)\big)\, e^{x}\, dx = f(x)\, e^{x} + C.$

Here the numerator and denominator are both degree 2, so first do a long division. Since $(x+1)^2 = x^2 + 2x + 1$,

$\frac{x^2 + 1}{x^2 + 2x + 1} = 1 + \frac{-2x}{(x+1)^2}.$

So the integral becomes

$I = \int e^{x}\, dx - 2\int \frac{x}{(x+1)^2}\, e^{x}\, dx = e^{x} - 2\int \frac{x}{(x+1)^2}\, e^{x}\, dx.$

Writing the second piece in derivative form

Add and subtract $1$ in the numerator: $x = (x+1) - 1$, so

$\frac{x}{(x+1)^2} = \frac{1}{x+1} - \frac{1}{(x+1)^2}.$

Now take $f(x) = \dfrac{1}{x+1}$. Then $f'(x) = -\dfrac{1}{(x+1)^2}$, which is exactly the $f(x) + f'(x)$ pattern, so

$\int \left( \frac{1}{x+1} - \frac{1}{(x+1)^2} \right) e^{x}\, dx = \frac{e^{x}}{x+1}.$

Putting it together,

$I = e^{x} - \frac{2e^{x}}{x+1} + C.$

Question 2: substitution to a logarithm

Evaluate

$I = \int \frac{3ax}{b^2 + c^2 x^2}\, dx.$

Let $t = b^2 + c^2 x^2$, so $dt = 2c^2 x\, dx$, which gives $x\, dx = \dfrac{1}{2c^2}\, dt$. Then

$I = \frac{3a}{2c^2} \int \frac{1}{t}\, dt = \frac{3a}{2c^2} \ln|t| + C = \frac{3a}{2c^2} \ln\left| b^2 + c^2 x^2 \right| + C.$

Question 3: integration by parts with tangent

Evaluate

$I = \int x\, \tan x\, \sec^2 x\, dx.$

Take the first function as $x$ and the second as $\tan x\, \sec^2 x$. For the second function, substitute $t = \tan x$, so $dt = \sec^2 x\, dx$ and

$\int \tan x\, \sec^2 x\, dx = \int t\, dt = \frac{t^2}{2} = \frac{\tan^2 x}{2}.$

Applying integration by parts,

$I = x \cdot \frac{\tan^2 x}{2} - \frac{1}{2}\int \tan^2 x\, dx.$

Using $\tan^2 x = \sec^2 x - 1$,

$\int \tan^2 x\, dx = \tan x - x.$

Therefore

$I = \frac{x}{2}\tan^2 x - \frac{1}{2}\tan x + \frac{x}{2} + C.$

Key takeaways

• If $e^{x}$ multiplies a fraction, try to reach the form $\int (f + f')e^{x}\, dx = f\, e^{x} + C$, dividing first when the degrees match.
• A factor of $x$ next to a quadratic denominator often signals the substitution $t =$ (the quadratic), leading to a logarithm.
• Integration by parts with $x$ as the first function reduces the problem to a simpler integral you can finish with a trig identity.