Indefinite Integrals and Substitution

Part 1: Basic Indefinite Integrals

Find the indefinite integrals for the following functions.

(a) $$\int (3e^{4x} - 5e^{-6x}) , dx$$

Solution:

$\int 3e^{4x} , dx = \frac{3}{4}e^{4x} + C$
$\int -5e^{-6x} , dx = \frac{5}{6}e^{-6x} + C$
Answer: $\frac{3}{4}e^{4x} + \frac{5}{6}e^{-6x} + C$

(b) $$\int (6x^6 + 4x^3 + 8) , dx$$

Solution:

$\int 6x^6 , dx = \frac{6}{7}x^7$
$\int 4x^3 , dx = x^4$
$\int 8 , dx = 8x$
Answer: $\frac{6}{7}x^7 + x^4 + 8x + C$

Solution:

$\int e^y , dy = e^y$
$\int \sin y , dy = -\cos y$
Answer: $e^y - \cos y + C$

(d) $$\int 4a^2 e^x , dx$$

Solution:

Treat $4a^2$ as constant.
$\int 4a^2 e^x , dx = 4a^2 e^x + C$

(e) $$\int (2 \tan x + 3 \sec x) , dx$$

Solution:

$\int 2 \tan x , dx = -2 \ln |\cos x|$ or $2 \ln |\sec x|$
$\int 3 \sec x , dx = 3 \ln |\sec x + \tan x|$
Answer: $2 \ln |\sec x| + 3 \ln |\sec x + \tan x| + C$

(f) $$\int \frac{3x^3 - 5x}{x^7} , dx$$

Solution:

Simplify: $\frac{3x^3}{x^7} - \frac{5x}{x^7} = 3x^{-4} - 5x^{-6}$
$\int 3x^{-4} , dx = -x^{-3}$
$\int -5x^{-6} , dx = x^{-5}$
Answer: $-x^{-3} + x^{-5} + C$

(g) $$\int (4 + 2x) , dx$$

Solution:

$\int 4 , dx = 4x$
$\int 2x , dx = x^2$
Answer: $4x + x^2 + C$

(h) $$\int 4e^4 , dx$$

Solution:

$4e^4$ is constant.
$\int 4e^4 , dx = 4e^4 x + C$

(i) $$\int (x + \sec x \tan x) , dx$$

Solution:

$\int x , dx = \frac{1}{2}x^2$
$\int \sec x \tan x , dx = \sec x$
Answer: $\frac{1}{2}x^2 + \sec x + C$

(j) $$\int \frac{\sin^2 x + \cos^2 x}{3x} , dx$$

Solution:

$\sin^2 x + \cos^2 x = 1$
$\int \frac{1}{3x} , dx = \frac{1}{3} \ln |x|$
Answer: $\frac{1}{3} \ln |x| + C$

(k) $$\int (x e^{-2x} - 3x) e^x , dx$$

Solution:

Simplify: $(x e^{-2x} - 3x) e^x = x e^{-x} - 3x e^x$
$\int x e^{-x} , dx$ requires integration by parts (not covered here).
Or combine: $\int (x e^{-x} - 3x e^x) , dx$
Answer: (Incomplete—requires integration by parts)

(l) $$\int 57x , dx$$

Solution:

$\int 57x , dx = \frac{57}{2} x^2 + C$

Part 2: Substitution Method

Use appropriate substitution to find the following.

(a) $$\int 3x (x^3 - 2)^3 , dx$$

Solution:

Let $u = x^3 - 2$, then $du = 3x^2 , dx$. However, integrand has $3x$, not $3x^2$. Adjust.
Alternatively, $du = 3x^2 dx \implies dx = \frac{du}{3x^2}$. Not directly matching.
Try $u = x^3 - 2$, $du = 3x^2 dx \implies \frac{du}{x} = 3x dx \implies 3x dx = \frac{du}{x}$. Still not clean.
Better: $u = x^3 - 2$, then $du = 3x^2 dx \implies 3x dx = \frac{du}{x}$. Not constant.
Answer: Requires adjustment or integration by parts.

(b) $$\int \cos x \sin^2 x , dx$$

Solution:

Let $u = \sin x$, then $du = \cos x , dx$.
Integral becomes $\int u^2 , du = \frac{u^3}{3} + C$
Answer: $\frac{1}{3} \sin^3 x + C$

Solution:

Let $u = x + 1$, then $du = dx$, $x = u - 1$.
Integral: $\int (u - 1) \sqrt{u} , du = \int u^{3/2} - u^{1/2} , du$
$= \frac{2}{5} u^{5/2} - \frac{2}{3} u^{3/2} + C$
Answer: $\frac{2}{5} (x+1)^{5/2} - \frac{2}{3} (x+1)^{3/2} + C$

(d) $$\int \mu e^{(1 + 1/\mu)} , d\mu$$

Solution:

Let $u = 1 + \frac{1}{\mu}$, then $du = -\frac{1}{\mu^2} d\mu$.
Then $\mu d\mu = - \mu^2 du$. Not directly matching.
Alternatively, set $u = \frac{1}{\mu}$, then $du = -\frac{1}{\mu^2} d\mu$.
$\mu e^{1 + 1/\mu} d\mu = - e^{1+u} \frac{1}{u^3} du$. Not simple.
Answer: Requires further manipulation.

(e) $$\int \frac{\sec^2 x}{(1 + \tan x)^3} , dx$$

Solution:

Let $u = 1 + \tan x$, then $du = \sec^2 x , dx$.
Integral: $\int u^{-3} , du = -\frac{1}{2u^2} + C$
Answer: $-\frac{1}{2(1 + \tan x)^2} + C$

(f) $$\int x \tan(x^2) \sec(x^2) , dx$$

Solution:

Let $u = x^2$, then $du = 2x , dx \implies x , dx = \frac{du}{2}$.
Integral: $\frac{1}{2} \int \tan u \sec u , du = \frac{1}{2} \sec u + C$
Answer: $\frac{1}{2} \sec(x^2) + C$

(g) $$\int \frac{t^2 + 2t}{t^7 + 3t^3 + 10} , dt$$

Solution:

Let $u = t^7 + 3t^3 + 10$, then $du = (7t^6 + 9t^2) dt$. Not matching numerator.
Try $u = t^3$, then $du = 3t^2 dt$, numerator becomes $t^2 + 2t$ which doesn't align.
Answer: Requires different substitution; not straightforward.

(h) $$\int \frac{(\ln x)^5}{x} , dx$$

Solution:

Let $u = \ln x$, then $du = \frac{1}{x} dx$.
Integral: $\int u^5 , du = \frac{u^6}{6} +

Source: Soalan Tut 1 FAD1014(25.26)..pdf