# Plus Two Maths Chapter Wise Previous Questions Chapter 7 Integrals

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 7 Integrals.

## Kerala Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 7 Integrals

### Plus Two Maths Application of Derivatives 3 Marks Important Questions

Question 1.
Find the following integrals. (May -2011)
$$\begin{array}{l} \text { (i) } \int x^{2} e^{2 x} d x \\ \text { (ii) } \int e^{x} \sin x d x \end{array}$$

Question 2.
(i) $$\int e^{x} \sec x(1+\tan x) d x=\ldots \ldots$$
(a) ex cosx + c (b) ex sec x + c
(C) ex tanx + c (d) ex sin x + c
(ii) Find $$\int \sin 2 x \cos 3 x d x$$ (March – 2014)

Question 3.
Find the following integrals.
(i) $$\begin{array}{l} \text { (i) } \int \frac{1}{(x+1)(x+2)} d x \\ \text { (ii) } \int \frac{2 x-1}{(x-1)(x+2)^{2}} d x \end{array}$$ (March – 2014)

### Plus Two Maths Application of Derivatives 4 Marks Important Questions

Question 1.
Consider the integral $$I=\int_{0}^{\pi} \frac{x \sin x}{1+\cos ^{2} x} d x$$
(i) Express $$I=\frac{\pi}{2} \int_{0}^{\pi} \frac{\sin x}{1+\cos ^{2} x} d x$$
(ii) Show that $$I=\frac{\pi^{2}}{4}$$ (March – 2012)

Question 2.
(i) Evaluate: $$\int_{2}^{3} \frac{x}{x^{2}+1} d x$$
(ii) Evaluate: $$\int_{0}^{\pi} \frac{x}{1+\sin x} d x$$ (March – 2014)

Question 3.
(a) What is the value of $$\int_{0}^{1} x(1-x)^{9} d x$$ If the
$$\begin{array}{llll} \text { (i) } \frac{1}{10} & \text { (ii) } \frac{1}{11} & \text { (iii) } \frac{1}{90} & \text { (iv) } \frac{1}{110} \end{array}$$
(b) Find $$\int_{0}^{1}(2 x+3) d x$$ of a sum. (March – 2015)

Question 4.
Evaluate $$\int_{0}^{x} \log (1+\cos x) d x$$

Question 5.
Find $$\int_{0}^{5}(x+1) d x \text { as limit of a sum. }$$

Question 6.
Evaluate $$\int_{0}^{4} x^{2} d x$$ as the limit of a sum. (March – 2017)

### Plus Two Maths Application of Derivatives 6 Marks Important Questions

Question 1.
(i) Fill in the blanks; $$\int \frac{1}{x} d x=$$_____
(ii) Evaluate $$\int \frac{5 x+1}{x^{2}-2 x-35} d x$$
(iii) Integrate with respect to x. $$\sqrt{x^{2}+4 x+8}$$ (March – 2010)

Question 2.
(i) Evaluate $$\int-\frac{\cos e c^{2} x}{\sqrt{\cot ^{2} x+9}} d x$$
(ii) Evaluate $$\int\left(\cos ^{-1} x\right)^{2} d x$$ (May -2010)

Question 3.
(i) Evaluate $$\int_{0}^{\pi} \frac{x \sin x}{1+\cos ^{2} x} d x$$
(ii) Evaluate $$\int_{0}^{2} e^{x} d x \text { as limit of a sum. }$$ (May -2010)

Question 4.
(i) Fill in the blanks $$\int \cot x d x=$$_____
(ii) Evaluate the integrals
$$\begin{array}{l} \text { (a) } \int \sin 2 x \cos 4 x d x \\ \text { (b) } \int \frac{x}{(x+1)(x+2)} d x \end{array}$$ (March -2011)

Question 5.
(i) Evaluate $$\int_{0}^{1} x d x$$ as the limit of a sum.
(ii) Evaluate $$\int_{0}^{1} x(1-x)^{n} d x$$ (March – 2011)

Question 6.
(i) Evaluate $$\int_{1}^{2} \frac{1}{x(1+\log x)^{2}} d x$$
(ii) Evaluate $$\int_{0}^{3}\left(2 x^{2}+3\right) d x$$ as the limit of a sum. (May – 2011)

Question 7.
(i) What is $$\int \frac{1}{9+x^{2}} d x=?$$
(ii) Evaluate the integrals $$\int \frac{1}{1+x+x^{2}+x^{3}} d x$$ (May – 2012)

Question 8.
(i) Evaluate $$\int_{0}^{3} f(x) d x$$
where $$f(x)=\left\{\begin{array}{ll} x+3, & 0 \leq x \leq 2 \\ 3 x, & 2 \leq x \leq 3 \end{array}\right.$$

(ii) Prove that $$\int_{0}^{1} \log \left(\frac{x}{1-x}\right) d x=\int_{0}^{1} \log \left(\frac{1-x}{x}\right) d x$$ Find the value of $$\int_{0}^{1} \log \left(\frac{x}{1-x}\right) d x$$ (May – 2012)

Question 9.
(i) Find $$\int \cot x d x=\ldots \ldots$$
(ii) Find $$\int x \log x d x$$
(iii) Find $$\int \frac{x-1}{(x-2)(x-3)} d x$$ (March – 2013)

Question 10.
Evaluate
$$\text { (i) } \int \frac{x+3}{\sqrt{5-4 x-x^{2}}} d x$$
$$\text { (ii) } \int_{\pi / 6}^{\pi / 3} \frac{d x}{1+\sqrt{\tan x}}$$ (May – 2013)

Question 11.
Evaluate
$$\begin{array}{l} \text { (i) } \int x^{2} \tan ^{-1} x d x \\ \text { (ii) } \int_{-1}^{2} x^{3}-x d x \end{array}$$ (May – 2013)

Question 12.
Evaluate $$\int_{0}^{\pi / 4} \log (1+\tan x) d x$$ (March – 2013)

Question 13.
(a) The value of $$\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \cos x d x$$ (May – 2014)
(b) Prove that $$\int_{0}^{\pi} \frac{x}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x} d x=\frac{\pi^{2}}{2 a b}$$

Question 14.
(a) $$\int \frac{1}{x^{2}+a^{2}} d x=$$
(b) Find $$\int \frac{1}{9 x^{2}+6 x+5} d x$$
(c) Find $$\int \frac{x}{(x-1)^{2}(x+2)} d x$$ (May – 2014)

Question 15.
Integrate the following
$$\begin{array}{l} \text { (a) } \frac{x-1}{x+1} \\ \text { (b) } \frac{\sin x}{\sin (x-a)} \\ \text { (c) } \frac{1}{\sqrt{3-2 x-x^{2}}} \end{array}$$ (March – 2015)

Question 16.
(a) Prove that $$\int \cos ^{2} x d x=\frac{x}{2}+\frac{\sin 2 x}{4}+c$$
(b)Find $$\int \frac{1}{\sqrt{2 x-x^{2}}} d x$$
(c) Find $$\int x \cos x d x$$ (May – 2015)
$$\begin{array}{l} \text { (i) } \int \frac{1}{x\left(x^{7}+1\right)} d x \\ \text { (ii) } \int_{1}^{4}|x-2| d x \end{array}$$
Find $$\int_{0}^{\frac{\pi}{2}} \log \sin x d x$$
Find the following: $$\begin{array}{l} \text { (i) } \int \cot x \log (\sin x) d x \\ \text { (ii) } \int \frac{1}{x^{2}+2 x+2} d x \\ \text { (iii) } \int x e^{9 x} d x \end{array}$$ (May – 2017)